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