How to solve $\int_0^{-2w}\phi^{-k-\frac{1}{2}}\frac{{_2F_1(k-\frac{1}{2},k,k+\frac{1}{2};1-\frac{k-\frac{3}{2}}{\phi})}}{\sqrt{-\phi-2\omega}}$?

Question

Please help me to solve the integral:

$$\int_0^{-2w}\phi^{-k-\frac{1}{2}}\frac{{_2F_1(k-\frac{1}{2},k,k+\frac{1}{2};1-\frac{k-\frac{3}{2}}{\phi})}}{\sqrt{-\phi-2\omega}}$$

I have solved this in Mathematica. But I am not able to way a general result even in the table of integrals.