Class of function which $\int_0^{\infty} x^{a-1} e^{-f(x)} dx$ is defined

Question

I am looking for a class of functions, which the following integral can be defined, and has a closed form value. $$ \int_0^{\infty} x^{a-1} e^{-f(x)} dx $$ where $a$ is an integer.

Quadratic functions have this property. But, do the higher powers have closed form?