Primitive of x by Root of a x squared plus b x plus c

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

Then:
 * $\ds \int x \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac {b \paren {2 a x + b} \sqrt {a x^2 + b x + c} } {8 a^2} - \frac {b \paren {4 a c - b^2} } {16 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$

Also see

 * Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$


 * For $a = 0$, see Primitive of $x \sqrt {a x + b}$