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

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} } = \begin {cases}

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