I am trying to integrate the following
$\int_{0}^{1} {_2{F}_1}\big(-n,1+2m+n,1+m,1-z\big)^2 dz$,
where $m\in\Bbb Z$ and $n\in\Bbb Z$ with $m>0$, $n\geq 0$.
(Basically I want to normalise the function). I am getting this as a part of an effort to find some uniform large approximation for a certain associated Legendre polynomial (in case people are wondering whether it is homework).
I know that
$\int_{0}^{1} (z(1-z))^m {_2{F}_1}\big(-n,1+2m+n,1+m,1-z\big)^2 dz$,
is integrable, under the same conditions, and gives
$\frac{n! \Gamma (m+1)^2 \Gamma (m+n+1) \Gamma (2 m+2 n+1)}{2\Gamma (m+n+1) \Gamma (2 m+n+1) \Gamma (2 m+2 n+2)}\quad\;{_2{F}_1}\big(0, 1 + 2 m + 2 n, 2 m + 2 + 2 n, 1\big)$.
The above is a table integral from I. S. Gradshteyn and I. M. Ryzhik 8th edition (doesn't appear in the 7th)
I welcome any hints or suggestions.