If I have for any piecewise continuous function $f$ $$ F(x) = \int f(x)\ dx $$ is it possible to calculate $$ \int f(ax+b)\ e^{cx}\ dx $$ If $f$ is $sin$ then a computer algebra system gives me $$ (e^{cx}\ (sin(ax + b) - a\ cos(ax + b)))/(a^2 + c^2) $$ However for a general $f$ the system says that there is no solution using standard mathematical functions.
I am not necessarily looking for an actual formula for it and argument for its existence would suffice (with not too onerous preconditions if required).