\[\int_{0}^{1}x^{\sigma}\left(1-x^{2}\right)^{\mu/2}\mathsf{P}^{-\mu}_{\nu}\left%
(x\right)\,\mathrm{d}x=\frac{\Gamma\left(\frac{1}{2}\sigma+\frac{1}{2}\right)%
\Gamma\left(\frac{1}{2}\sigma+1\right)}{2^{\mu+1}\Gamma\left(\frac{1}{2}\sigma%
-\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}\sigma+\frac{1}{%
2}\nu+\frac{1}{2}\mu+\frac{3}{2}\right)},\]