In calculating the multipole moments on the Maclaurin spheroid, I've encountered the following integral. The text I'm reading (Poisson and Will) says that the integral is doable.
$$ \int_{-1}^{1} \left(1+a x^2\right)^{-\frac{l+3}{2}} P_l(x) d x$$
The constant $a = e^2/\left(1-e^2\right)$ is a function of the spheroid's eccentricity. I'm trying to evaluate it for general $l\in\mathbb{Z}$, but for specific values of $l$ it gives me the correct result (so I'm certain this is the correct integral to evaluate).
I've tried integrating by parts with $P_l(x)\propto \frac{d^l}{dx^l}\left(x^2-1\right)^l$, but the result is just as difficult to crack because then I have $l$ derivatives of the first term in the integrand and WolframAlpha tells me this is a hypergeometric function.
