I obtained the following integral when looking for a probability density function: $$\int_0^1 x^{\alpha1} \,(1x) ^{A}\, {}_2F_1 (1A, \alpha 1A, \alpha A, x) \,dx$$ Can anyone please give me some hints of evaluating the value of the integral?

If $\Re(A)<1$ and $\Re(\alpha)>0$ Mathematica gives: $$\int_0^1 \mbox{ }x^{\alpha1} \,(1x) ^{A}\, {}_2F_1 (1A, \alpha 1A; \alpha A; x) \mbox{ }dx$$ $$\mbox{ }=\frac{\Gamma(\alpha)\Gamma(1A)}{\Gamma(1+\alphaA)}\mbox{ } \mbox{}_3F_2(\alpha,1A,\alphaA1;\alphaA,1+\alphaA;1)$$ 

