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

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

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

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.