Primitive of Reciprocal of x squared by square of a x squared plus b x plus c/Proof 1

Theorem
Let $a \in \R_{\ne 0}$.

Then:
 * $\displaystyle \int \frac {\mathrm d x} {x^2 \left({a x^2 + b x + c}\right)^2} = \frac {-1} {c x \left({a x^2 + b x + c}\right)} - \frac {3 a} c \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2} - \frac {2 b} c \int \frac {\mathrm d x} {x \left({a x^2 + b x + c}\right)^2}$

Proof
From Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c:

Setting $m = n = 2$: