Why $\int \frac{\ln(x)^n}{(x-s)(x-s_0)} dx \propto \ln(x)^{n+1}$

Question

I am currently trying to find an argument for the fact

$\int \frac{\ln(x)^n}{(x-s)(x-s_0)} dx \propto \ln(x)^{n+1}$

This seems to be approximatively the case from looking at the first few powers of the logarithm calculated with Wolfram. There always is a term proportional to $\ln(x)^n\ln(1-\frac{x}{s})$ and then there is a whole mess of polylogarithmic functions. My question is whether there is a simple to way to extract this leading logarithm behavior.

Answer 1

Hints: $$\int\frac{\log^n x}{x}\,dx = \frac{1}{n+1}\log^{n+1} x,\qquad \frac{1}{(x-s_0)(x-s_1)}=\frac{1}{s_0-s_1}\left(\frac{1}{x-s_0}-\frac{1}{x-s_1}\right).$$