Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0

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

Then for $x \in \R$ such that $a x^2 + b x + c > 0$:


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

\dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\ \dfrac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \dfrac 1 {\sqrt a} \ln \size {2 a x + b} + C & : b^2 - 4 a c = 0 \end {cases}$ where $\arsinh$ denotes the area hyperbolic sine function.