How to solve this integral?
$\displaystyle \int \frac{r \: dr}{\sqrt{r^4 l^{-2}+ r^2(1 + a^2 l^{-2}) - 2 m r + a^2}}$
where, $a,l,m \in \mathbb{R}$ and $m > 0$.
I tried putting this in Mathematica for some parameter values, and it gave some complicated function of elliptic integrals.
$$I=\int\frac{rdr}{\sqrt{c_1r^4+c_2r^2+c_3r+c_4}}$$ where: $c_1=l^{-2}$, $c_2=1+a^2l^{-2}$, $c_3=-2m$ and $c_4=a^2$
This would be my first try but unlike a quadratic, this quartic does not seem to have a general solution