On Some Integrals Over the Product of Three Legendre Functions

The definite integrals $\int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^3\D x$, $\int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^2P_{\nu}(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^3\D x$ and $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^2P_{\nu+1}(-x)\D x$ are evaluated in closed form, where $P_\nu$ is the Legendre function of degree $\nu$, and $\R\nu>-1$. Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.