Calculate $\int \frac{r \: dr}{\sqrt{r^4 l^{-2}+ r^2(1 + a^2 l^{-2}) - 2 m r + a^2}}$

Question

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.

Are you trying to find out how to derive the complicated function of elliptic integrals, or are you trying to find a simpler solution? If Mathematica gives you elliptic integrals, it is quite likely there is (with the exception of some specific values of your parameters) no simpler solution. — Kusma, Sep 09 '18 at 19:11
An elliptic integral is what it is, Rahul. An efficient approach for the numerical evaluation of complete elliptic integrals of the first kind is the AGM mean. Also, the parameters $a,l,m$ are pretty irrelevant here. — Jack D'Aurizio, Sep 09 '18 at 19:33
Regarding whether there exists an antiderivative in terms of elementary functions, see When is an elliptic integral expressible in terms of elementary functions?. Also, for what it's worth, I spent an entire two weeks at the end of April 2018 playing around with one particular elliptic integral here. — Dave L. Renfro, Sep 12 '18 at 16:03

score 0 · Answer 1 · answered Sep 12 '18 at 14:01

0

$$I=\int\frac{rdr}{\sqrt{c_1r^4+c_2r^2+c_3r+c_4}}$$ 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

answered Sep 12 '18 at 14:01

Henry Lee

1 Answers1