A definite integral contianing ln(x)

Question

everyone, I met a tough definite integral as follows, $$I = \int\limits_1^\infty {\frac{{\ln x}}{{{{\left( {x + a} \right)}^m}{{\left( {x + b} \right)}^{n + 1}}}}} dx,$$ where $a$ and $b$ are constant, $m$ and $n$ are non-negative integers.

Answer 1

If $~m,n\in\mathbb N,~$ then our integral can be expressed in terms of $~J^{(m-1,~n)}(a,b),~$ where $J(a,b)=$

$$=\int_1^\infty\frac{\ln x}{(x+a)(x+b)}~dx~=~\int_0^\infty-\int_0^1~=~\frac{\dfrac{\ln^2a-\ln^2b}2+\text{Li}_2\bigg(-\dfrac1a\bigg)-\text{Li}_2\bigg(-\dfrac1b\bigg)}{a-b},$$

see dilogarithm $($ or Spence's function $)$ and polylogarithm for more details.