I am trying to model the behavior of a certain phenomena and in the course of it I have got a complicated integral which involves among other things an unknown function $f(x)$ (which I abbreviate to $f$). I have the option of choosing this function $f$ so that the integral can be evaluated easily.
Specifically, let $g(x)=(2^{36/5}(-5x-1)^{2/5}(x+1)^2)/x$ and consider the indefinite integral $\int \big (\frac{2f(1+x)}{1+5x}-\frac{1+x}{x(1+5x)}\big )g(x)dx$. I want to choose the function $f$ so that $f(0)=0$, $f$ is increasing and that the integral takes a particularly simple form in terms of elementary functions. How do I do that? I have been randomly selecting functions $f$ and trying them out but to no avail. Is there a standard way to solve such a problem?
Thanks for your time.
Edit: If I let the integrand be denoted by $F$, then $f$ evaluates to $\frac{Fx(1+5x)^{3/5}}{2^{41/5}(1+x)^3}+\frac{1}{2x}$. But now there is no way I can have f(0)=0. How do I get over this problem?