What are those functions

Question

what are such functions $$f(x) = \int_{c}^{x} R \left(t, (P(t))^{\frac{1}{n}} \right) \, dt,$$ where $R$ is a rational function of its two arguments, P is a polynomial with no repeated roots, $n \geq 3$and $c$ is a constant.Any research on such a kind of function?And what is the inverse function of $f(x)$?Since we know $n=2$,it is elliptic integral,and it's inverse function is elliptic function.

Answer 1

It's a special case of an (incomplete) Abelian integral. Namely, we have the integral $$\int_c^x R(t,y)_{f=0}\, dt$$ where we require that $(y,t)$ lie on the algebraic curve $f(t,y)=y^n-P(t)=0$.