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

Lemma for Primitive of $\dfrac 1 {x^2 \paren {a x + b}^3}$

 * $\dfrac 1 {x^2 \paren {a x + b}^3} \equiv \dfrac {-3 a} {b^4 x} + \dfrac 1 {b^3 x^2} + \dfrac {3 a^2} {b^4 \paren {a x + b} } + \dfrac {2 a^2} {b^3 \paren {a x + b}^2} + \dfrac {a^2} {b^2 \paren {a x + b}^3}$

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

Equating constants in $(1)$:

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

Equating $4$th powers of $x$ in $(1)$:

Equating $3$rd powers of $x$ in $(1)$:

Summarising:

Hence the result.