Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Proof 2

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

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

\dfrac {-1} {\sqrt c} \ln \left({\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x}\right) & : b^2 - 4 a c > 0 \\ \dfrac {-1} {\sqrt c} \sinh^{-1} \left({\dfrac {b x + 2 c} {\left\vert{x}\right\vert \sqrt {4 a c - b^2} } }\right) & : b^2 - 4 a c < 0 \\ \dfrac {-1} {\sqrt c} \ln \left\vert{\dfrac {2 c} x + b}\right\vert + C & : b^2 - 4 a c = 0 \end{cases}$

Proof
Then:

Now provided $N$ is real and non zero: