Is it known in the literature how the following integral can be evaluated analytically?
$$\int_0^a dx\frac{\Gamma(1+x)}{\Gamma(1+x+a)}x^n=???$$
where $a>0$ and $n\in\mathbb{Z}^*$.
I've tried using series representations and integral representations of Γ(z) and 1/Γ(z) to first obtain a simple integral in x and then consider resumming the expansions. That did not help much. Additionally, Gradshteyn and Ryzhik does not seem to contain the answer either.