I need to integrate a bunch off different functions on the form
$$\frac{x^\gamma}{(a+b x^2)^\alpha (c+d x^2)^\beta}$$
were the integrand is strictly positive and $\alpha$, $\beta$ are positive integers of half integers, and $\gamma$ is a positive integer. I understand (my playing with Mathematica) that integrals of this type explode into a barrage of terms.
Take for instance the (simple) example $$ F=\frac{1}{\left(a+b x^2\right)^2 \sqrt{c+d x^2}} $$ which integrates to $$ \frac{1}{2 a^{3/2}}\left(\frac{\sqrt{a} b x \sqrt{c+d x^2}}{\left(a+b x^2\right) (b c-a d)}+\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{(b c-a d)^{3/2}}\right).$$
Could someone please enlighten me as to the type of transformation that is at play here.