Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Positive Discriminant

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

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

Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $\size {2 a x + b} < \sqrt {b^2 - 4 a c}$:


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

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

Then:

Thus: