Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma

Theorem
Let $a, b, c \in \R_{\ne 0}$.

Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $x \ne 0$:
 * $\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\pm \sqrt {a + b u + c u^2} }$

where $u := \dfrac 1 x$, according to whether $u > 0$ or $u < 0$.