$\int x\sqrt{6x^2+36x+51}dx$

$\frac{\sqrt{\left(2\right)^{3}}\sqrt{3}\sqrt{\left(\left(x+3\right)^2-\frac{1}{2}\right)^{3}}}{6}+\frac{\sqrt{\left(3\right)^{3}}\ln\left|\sqrt{2}x+3\sqrt{2}+\sqrt{2}\sqrt{\left(x+3\right)^2-\frac{1}{2}}\right|}{\sqrt{2}}+\frac{\frac{-3\sqrt{3}}{\sqrt{2}}}{2}\ln\left|\sqrt{2}x+3\sqrt{2}+\sqrt{2}\sqrt{\left(x+3\right)^2-\frac{1}{2}}\right|+\frac{-3\sqrt{3}\sqrt{\left(x+3\right)^2-\frac{1}{2}}\left(x+3\right)^{2}}{\frac{\sqrt{\left(2\right)^{3}}}{2}\left(x+3\right)}+C_0$