A recent thread in the calculus forum reminded me of the following statement, which I've seen in a book:

If $\displaystyle m, n, p \in\mathbb{Q}$, then $\displaystyle \int x^m(a+bx^n)^p\;{dx}$ can be expressed in terms of elementary functions only when:

(a) $\displaystyle p$ is an integer, (b) $\displaystyle \frac{m+1}{n} $ is an integer, or (c) $\displaystyle \frac{m+1}{n}+p $ is an integer.My question: how is this proven?