$$\int\sqrt {\smash[b]{a-(b-x)^2}}\cdot \sqrt{\smash[b]{c-(d-x)^2}} \,\mathrm{d}x$$ Where $a$, $b$, $c$, $d$ are real numbers
I need to integrate this and I cannot think of a way to do it. I have contacted several people telling me it either concludes as elliptic integral or some nonstandard method.
Any help welcomed, thanks.
Notice that, for opportune values of $a, b, c, d$ your integral can be rewritten as
$$\int \sqrt{Ax^4 + Bx^3 + Cx^2 + Dx + E}\ \text{d}x$$
Where $A, B, C, D, E$ are constants defined by what you get in expanding the brakets and multiplying the square roots arguments (a legit process so far as $a, b, c, d$ are good, yet not a general statement).
In this case the integration results to be pretty grueling and tiresome. Of course it may happen that some of those coefficient might be zero, but a priori we cannot state anything for sure.
This is why I decided to run your initial integral on W. Mathematica, which provided the following cumbersome result. Yet true for all $a, b, c, d$.
$$\frac{-2 d \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right) b^3+2 \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} b^3-6 c \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right) b^2-2 d \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right) b^2+\left(d-x+\sqrt{c}\right) \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \sqrt{\frac{\sqrt{a} \left(d-x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(-b+x+\sqrt{a}\right) \left(-d+x+\sqrt{c}\right)-\left(b-x+\sqrt{a}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+d+\sqrt{a}-\sqrt{c}\right) E\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)+2 \left(b^2+\left(b+d+\sqrt{c}\right) \sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-4 \sqrt{a} (b+d) \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right)\right) b^2+4 d^2 \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} b^2-2 d^3 \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right) b-2 d^2 \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right) b-2 a \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right) b-2 d \left(d-x+\sqrt{c}\right) \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \sqrt{\frac{\sqrt{a} \left(d-x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(-b+x+\sqrt{a}\right) \left(-d+x+\sqrt{c}\right)-\left(b-x+\sqrt{a}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+d+\sqrt{a}-\sqrt{c}\right) E\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)+2 \left(b^2+\left(b+d+\sqrt{c}\right) \sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-4 \sqrt{a} (b+d) \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right)\right) b+2 a d \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} b+2 c d \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} b+10 c \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} b-6 a d^2 \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 c d \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right)+d^2 \left(d-x+\sqrt{c}\right) \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \sqrt{\frac{\sqrt{a} \left(d-x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(-b+x+\sqrt{a}\right) \left(-d+x+\sqrt{c}\right)-\left(b-x+\sqrt{a}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+d+\sqrt{a}-\sqrt{c}\right) E\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)+2 \left(b^2+\left(b+d+\sqrt{c}\right) \sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-4 \sqrt{a} (b+d) \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right)\right)+2 a \left(d-x+\sqrt{c}\right) \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \sqrt{\frac{\sqrt{a} \left(d-x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(-b+x+\sqrt{a}\right) \left(-d+x+\sqrt{c}\right)-\left(b-x+\sqrt{a}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+d+\sqrt{a}-\sqrt{c}\right) E\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)+2 \left(b^2+\left(b+d+\sqrt{c}\right) \sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-4 \sqrt{a} (b+d) \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right)\right)+2 c \left(d-x+\sqrt{c}\right) \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \sqrt{\frac{\sqrt{a} \left(d-x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(-b+x+\sqrt{a}\right) \left(-d+x+\sqrt{c}\right)-\left(b-x+\sqrt{a}\right) \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \left(\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+d+\sqrt{a}-\sqrt{c}\right) E\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)+2 \left(b^2+\left(b+d+\sqrt{c}\right) \sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-4 \sqrt{a} (b+d) \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right)\right)-\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(a-(b-x)^2\right) \left(c-(d-x)^2\right) (b+d-2 x) \sqrt{\frac{\sqrt{a} \left(d-x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}+8 a c \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}+2 d^3 \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}+10 a d \left(b-x+\sqrt{a}\right) \left(d-x+\sqrt{c}\right) \left(\left(b+\sqrt{a}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)-2 \sqrt{a} \Pi \left(\frac{-b+d+\sqrt{a}+\sqrt{c}}{-b+d-\sqrt{a}+\sqrt{c}};\sin ^{-1}\left(\sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}\right)|\frac{-b^2+2 d b-d^2+a+c+2 \sqrt{a} \sqrt{c}}{-b^2+2 d b-d^2+a+c-2 \sqrt{a} \sqrt{c}}\right)\right) \sqrt{-\frac{\sqrt{a} \left(-d+x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}-\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}} \sqrt{\frac{\left(b-d+\sqrt{a}-\sqrt{c}\right) \left(-b+x+\sqrt{a}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}}{6 \left(b-d+\sqrt{a}-\sqrt{c}\right) \sqrt{a-(b-x)^2} \sqrt{c-(d-x)^2} \sqrt{\frac{\sqrt{a} \left(d-x+\sqrt{c}\right)}{\left(-b+d+\sqrt{a}+\sqrt{c}\right) \left(b-x+\sqrt{a}\right)}}}$$
Where as you can see, Elliptic Function do appear. They are denoted by $F(\ldots)$.