Primitive of x by Power of a x + b

Theorem

 * $\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$

where $n \ne - 1$ and $n \ne - 2$.

Proof
Let $u = a x + b$.

Then:

Then:

Also see

 * Primitive of $x$ over $a x + b$ for the case when $n = -1$
 * Primitive of $x$ over $\paren {a x + b}^2$ for the case when $n = -2$