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

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

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

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

Equating constants in $(1)$:

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

Equating $2$nd powers of $x$ in $(1)$:

Summarising:

Hence the result.