p is a real number.Is $p$ an integer? If so, then you could do $u=\ln(x)$, which leads to
which succumbs to by-parts as many times as you need. I'd recommend tabular integration in that case. If $p$ can be real, you get something a bit more nasty, with the incomplete Gamma function in there. I don't know how you get that result. I'd have to study a bit.