This solution for the inverse of this function is g(x)=x-LambertW(e^x+1)+1. However there is a way around
y=e^x+ax+b
y-b=e^x+ax=Y
(Y-1)'=Int(1/Y')dx
By computing the integral above and reverting to original variable you will get
g(x)=x/a-[ln(e^x+a)]/a+b
