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

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

Then:
 * $\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = \begin {cases}

\dfrac {-1} {\sqrt c} \ln \size {\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x} & : b^2 - 4 a c > 0 \\ \dfrac {-1} {\sqrt c} \map {\sinh^{-1} } {\dfrac {b x + 2 c} {\size x \sqrt {4 a c - b^2} } } & : b^2 - 4 a c < 0 \\ \dfrac {-1} {\sqrt c} \ln \size {\dfrac {2 c} x + b} + C & : b^2 - 4 a c = 0 \end {cases}$