Primitive of $x^{-\alpha} (1-x)^{-\beta}$, $\alpha,\beta \ge 1$

Question

Let be $\alpha,\beta \ge 1$. Is there a simple, "explicit" formula for the primitive of $f(x) = x^{-\alpha} (1-x)^{-\beta}$, $x\in (0,1)$?

Answer 1

If $\alpha$ is not an integer, it is $$\frac{x^{1-\alpha}}{1-\alpha} \; {\mbox{$_2$F$_1$}(\beta,1-\alpha;\,2-\alpha;\,x)}$$

Answer 2

You can use the Chebyshev criterion for binomial integrals.