Primitive of x over a x + b by p x + q/Partial Fraction Expansion

Lemma for Primitive of x over a x + b by p x + q

 * $\dfrac x {\left({a x + b}\right) \left({p x + q}\right)} \equiv \dfrac b {\left({b p - a q}\right) \left({a x + b}\right)} - \dfrac q {\left({b p - a q}\right) \left({p x + q}\right)}$

Proof
Setting $p x + q = 0$ in $(1)$:

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

Summarising:

Hence the result.