I have to compute the integral
$$ \int_x^1 dz \, z^\alpha (1-z)^\beta \; _2F_1(a_1,a_2;b_1;z) $$ where $x$ is some positive real number $0 \le x<1$. If $x=0$ then the answer is a $_3F_2$ (see here), but I don't know how to solve this for general $x$, and Mathematica gives no answer. Does anybody know if there is any closed form for this integral?
If that helps, what I'm interested is the case $a_1=a_2=b_1/2>0$, but general $\alpha$ and $\beta$ (real numbers such that the integral converges).