A general form for the indefinite integral of your problem does not exist.
Take y = exp(a*x) and transform the integral over x to an integral over y. It will be the integral of
1/a* 1/(1+b*y^(c/a)) *dy
depending on what the value of c/a is, a general form for the integral may/may not exist. For example, for c/a=1, the result is
but for c/a=2, b>0, the integral will be
So I don't think you can get a general form of the integral from the Symbolic Math Toolbox or any other Symbolic Math Software. You can use the numerical integrations methods and integrate it over a definite domain.