Integrating functions of the form $\Big(\frac{P(t)}{Q(t)}\Big)^{1/2}$ where $P$ and $Q$ are quadratic polynomials of degree $\leq 2$

Question

There is a general method for integrating any function of the form $\sqrt{P(t)}$ where $P$ is a polynomial of degree $\leq 2$. I was wondering if it possible to find

$$\int \sqrt{\frac{P(t)}{Q(t)}}dt$$

for polynomials $P$, $Q$ of degree $\leq 2$ using elementary functions.

Answer 1

For degree $1$, it is quite simple to compute $$I=\int \sqrt{\frac{x-a}{x-b}}\,dx$$ Let $$u=\sqrt{\frac{x-a}{x-b}}\implies x=\frac{b u^2-a}{u^2-1}\implies dx=\frac{2 u (a-b)}{\left(u^2-1\right)^2}\,du$$ This makes $$I=2(a-b)\int \frac{ u \sqrt{u^2} }{\left(u^2-1\right)^2}\,du=2(a-b)\int \frac{ u^2 }{\left(u^2-1\right)^2}\,du$$ Using partial fraction decomposition $$\frac{ u^2 }{\left(u^2-1\right)^2}=-\frac{1}{4 (u+1)}+\frac{1}{4 (u+1)^2}+\frac{1}{4 (u-1)}+\frac{1}{4 (u-1)^2}$$ which is simple.

All other situations lead to monsters in terms of elliptic integrals.