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

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

Let $b^2 - 4 a c < 0$.

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


 * $\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$

where $\arsinh$ denotes the area hyperbolic sine function.

Completing the Square
Let $b^2 - 4 a c < 0$.

Then:

Thus: