Consider the integral $$ \int \frac{(a+bx)^k}{x} dx. $$ Does this have a representation using (preferably) elementary or special functions?
Edit: $0 <k<1$.
Wolfram alpha produces an expression involving the (I did not know about this function beforehand) 2F1 hypergeometric function without explanation. I guess it can be attained by the integral representation here, but I am hoping something simpler might exist - in particular something which is easier to evaluate numerically.