For a project I want to get a closed form solution of $$\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$$ Here $p \in \mathbb{N},\; d\ge3, \; d\in\mathbb{N}$ and $P_n^d$ is the associated legendre polynomial.
The "Table of Integrals, Series and Products by Gradshteyn and Ryzhik" has several related integrals in $\S7.12-\S7.13$, for example $\S7.132.5$ gives a closed form for $\int_{0}^1 x^p (1-x^2)^{d/2}P_n^d(x) dx$ which seems to be quite close but unfortunately I am not fluent enough in Legendre polynomials to quickly figure out the modifications that i'd need to do. I'd be happy if anyone could give any pointers, Thanks.