\[\int\frac{x}{\left(1-x^{2}\right)^{3/2}}\mathsf{P}^{\mu}_{\nu}\left(x\right)%
\mathsf{Q}^{\mu}_{\nu}\left(x\right)\,\mathrm{d}x=\frac{1}{\left(1-4\mu^{2}%
\right)\left(1-x^{2}\right)^{1/2}}\left((1-2\mu^{2}+2\nu(\nu+1))\mathsf{P}^{%
\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)+(2\nu+1)(\mu-\nu-%
1)x(\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)%
+\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right))+2%
(\mu-\nu-1)^{2}\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}%
\left(x\right)\right),\]