Solving an elliptic integral so at the moment I am stuck at an integral of this form, maybe someone could help me.
$$\int \frac{dr}{r^{2}\sqrt{c+\frac{b}{r^{2}}+ar^{2}}}$$

Not an elliptic integral. Assuming the integration range is a subset of $\mathbb{R}^+$ and the polynomial $b+cr^2+ar^4$ has no positive root,
$$\int \frac{dr}{r\sqrt{b+cr^2+a r^4}}\stackrel{r\mapsto\sqrt{u}}{=}\frac{1}{2}\int\frac{du}{u\sqrt{b+cu+au^2}}. $$
By a suitable affine substitution the problem boils down to computing
$$ \int \frac{du}{(Au+B)\sqrt{u^2+1}}\stackrel{u\mapsto\sinh t}{=}\int\frac{dt}{A\sinh t+B} $$
and the further substitution $t=\log z$ turns the last integral into an elementary integral, which can be tackled through partial fraction decomposition.
