Reduction Formula for Primitive of Product of Power with Power of Quadratic

Theorem
Let $n \in \Z_{\ge 0}$ and $k \in \Z_{\ge 2}$.

Let:
 * $I_{n, k} := \ds \int x^k \paren {x^2 + A x + B}^n \rd x$

Then:
 * $I_{n, k} = \dfrac {x^{k - 1} \paren {x^2 + A x + B}^{n + 1} } {k + 2 n + 1} - \dfrac {B \paren {k - 1} } {k + 2 n + 1} I_{n, k - 2} - \dfrac {A \paren {k + n} } {k + 2 n + 1} I_{n, k - 1}$

is a reduction formula for $\ds \int x^k \paren {x^2 + A x + B}^n \rd x$.

Proof
With a view to expressing $I_{n, k}$ in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then: