I am wondering whether there exists a closed form for the definite integral $$F(x)=\int_0^1t^{-a}(1-t)^{N}(1-xt)^{-a}{}_2F_1(-a,k-a-1/2,k-a;4xt(1-xt))dt,$$ where $a\in(0,1)$ and $N,k$ are positive integers.

It seems that I cannot apply Kummer's quadratic transformation formula directly. So I tried to follow the method of proving Kummer's formula by expanding $_2F_1$, and I require some formula for $_2F_1(A,-A,C;x)$. However I cannot find anything relevant in many literatures.

Do you have some other ideas?

Thanks in advance.