$$\begin{aligned} \int \frac{x}{3 x^{2}+x+2}=\frac{1}{6} \int \frac{6 x+1-1}{3 x^{3}+x+2} &=\frac{1}{6} \int \frac{6 x+1}{3 x^{2}+x+2}-\frac{1}{6} \int \frac{1}{3 x^{2}+x+2} \\ &=\frac{1}{6} \ln \left|3 x^{2}+x+2\right|-\frac{1}{6} \int \frac{1}{3\left(x+\frac{1}{6}\right)^{1}+\frac{23}{12}} \\ &=\frac{1}{6} \ln \left|3 x^{2}+x+2\right|+\frac{\sqrt{12}}{18} \arctan \left(\frac{x+\frac{1}{6}}{\sqrt{\frac{23}{12}}}\right) \end{aligned}$$
