Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0/Proof 2

Theorem
Let $a, b, p, q \in \R$ such that $a p \ne b q$.

Proof
First let us express the integrand in the following form:

Recall the definition of Euler's third substitution:

In this context we have:

Hence we make the substitution:

Then:

Then:

Assembling the pieces: