Primitive of Reciprocal of x squared by a x + b squared/Partial Fraction Expansion

Lemma for Primitive of Reciprocal of x squared by a x + b squared

 * $\dfrac 1 {x^2 \left({a x + b}\right)^2} \equiv -\dfrac {2 a} {b^3 x} + \dfrac 1 {b^2 x^2} + \dfrac {2 a^2} {b^3 \left({a x + b}\right)} + \dfrac {a^2} {b^2 \left({a x + b}\right)^2}$

Proof
Setting $a x + b = 0$ in $(1)$:

Equating constants in $(1)$:

Equating $1$st powers of $x$ in $(1)$:

Equating $3$rd powers of $x$:

Summarising:

Hence the result.