I am currently struggling to find out where the following equation comes from. The authors of the article (Bayer, Friz, Gatheral: Pricing under rough volatility, p.12) where I got it from just wrote it down like it is obvious. Let $\gamma\in(0,\frac{1}{2})$ and $x>1$. Then
$\int\limits_0^1(1-s)^{-\gamma}(x-s)^{-\gamma}ds=\frac{x^\gamma}{1-\gamma}\text{ }_2F_1(1,\gamma,2-\gamma,x)$
where $_2F_1$ denotes the hypergeometric function.
I think you mean to integrate from $0$ to $x$, and you also got the exponent of $x$ wrong. Maple says $$ \int_0^x (1-s)^{-\gamma} (x-s)^{-\gamma} ds = \frac{x^{1-\gamma}}{1-\gamma} {}_2F_1(1,\gamma; 2-\gamma; x)$$ which it apparently gets using the "meijerg" method.