My goal is to compute the following integral, $$\int_{-1}^{1}P^\mu_\nu(x)(1-x^2)^{\mu/2+1/2}dx.$$ I tried to compute this using Wolfram Alpha and Maple but unfortunately did not get any result. Furthermore, I found a similar identity here https://dlmf.nist.gov/14.17 but the power on the term $(1-x^2)$ is not the same as in my case. Any references or hints on how to compute this expression will be much appreciated.
The parameters $\mu,\nu \in \mathbb{R}.$