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

Theorem
Let $a, b, c \in \R$ such that $a > 0$.

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

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


 * $\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C$

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

Then:

Thus: