General form for the integral of $ab\over c$

Question

Given $a(x)$, $b(x)$, and $c(x)$, assuming I know the first derivatives and integrals of each, is there a general form for $\int {ab\over c} dx$?

Answer 1

In general there is no simple function that will give out an antiderivative of $\frac{ab}{c}$ in terms of the derivatives and anti-derivatives of $a$, $b$, $c$.

As an example, it is well-known that there is no closed form for the anti-derivative of the function $f(x)=e^{-x^2}$ (related to the Normal density).

Yet away from $0$ this function can be written $$f(x)=\frac{a(x)b(x)}{c(x)}$$ where $$\cases{a(x)=xe^{-x^2}\\b(x)=1\\c(x)=x}$$ and all the derivatives and anti-derivatives of $a,b,c$ have closed forms.