when $\int x^m (a+bx^n)^p dx$ is elementary function

1.3k Views Asked by Bumbble Comm At 26 Mar 2026 - 5:30

In one of the answers of the question Integration of sqrt Sin x dx, I saw something similiar to that:

$m,n,p \neq 0 \in \mathbb{Q}$

$\int x^m (a+bx^n)^p dx$ is elementary function $\implies$ $\frac{m+1}{n} \in \mathbb{Z}$ or $\frac{m+1}{n} + p \in \mathbb{Z}$

is it true?

the other direction also true?

and where can I find a proof of this proposition?

There are 1 best solutions below

Bumbble Comm On 17 Jan 2016 - 9:57 BEST ANSWER

This is a result of Chebyshev.

A Google search for "chebychev elementary functions integration" turns up a number of relevant hits including these:

Also look up "integration in finite terms".