I was trying to see if there is a simple way to compute the following integral, where $0, $ \int_a^b\log(x)\frac{\sqrt{(x-a)(b-x)}}{x(1-x)}dx. $ Any idea ?
NB : With change of variables and the use of series expansion of $\log(1+x)$, one can reduce the problem to compute for all $k\geq 0$ $ \int_0^{b-a}y^k\frac{\sqrt{y(b-a-y)}}{(y+a)(1-a-y)}dy, $ but then I'm stuck ...