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

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

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

Then:


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

is not defined.

Proof
Suppose that $b^2 - 4 a c = 0$.

Then:

But we have that:
 * $\paren {2 a x + b}^2 > 0$

while under our assertion that $a < 0$:
 * $4 a < 0$

and so:
 * $a x^2 + b x + c < 0$

Thus on the real numbers $\sqrt {a x^2 + b x + c}$ is not defined.

Hence it follows that:
 * $\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$

is not defined.