\int x \sqrt{a x^2 + bx + c} = \frac{1}{48a^{5/2}}\left ( 
2 \sqrt{a} \sqrt{ax^2+bx+c} - \left( 3b^2 + 2 abx + 8 a(c+ax^2) \right)
\right.
\\
\left. + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+x} \right| \right)