Integral $\int{\frac{5x}{(2x^2-3) \sqrt{3x^2-2x+1}}}dx$ I need to solve the following problem:
$$\int{\frac{5x}{(2x^2-3) \sqrt{3x^2-2x+1}}}dx$$
I tried with trigonometric substitution, but I couldn't solve it. The square root is messing me up. I searched it up and nothing appears to work well so I hope someone can help. Thanks.

Integrate as follows\begin{align}
& \int{\frac{x}{(2x^2-3) \sqrt{3x^2-2x+1}}}\ dx\\
=& \ \frac12\int \left(\frac{1}{2 x+\sqrt{6}}+\frac{1}{2 x-\sqrt{6}}\right)\frac1{\sqrt{3
   x^2-2 x+1} }\ dx\>\>\>\>\>\>\>t_{\pm}=\frac1{2x\pm \sqrt6}\\
=& -\frac14\int \frac {1}{\sqrt{3-2(2+3\sqrt6)t_+ +2(11+2\sqrt6)t_+^2}}\ dt_+ \\
&\>-\frac14\int \frac {1}{\sqrt{3-2(2-3\sqrt6)t_- +2(11-2\sqrt6)t_-^2}}\ dt_-\\\
=& \ - \frac{\sinh^{-1}\frac{2(11+2\sqrt6)t_+-(2+3\sqrt6)}{2\sqrt2}}{4\sqrt{2(11+2\sqrt6)}}- \frac{\sinh^{-1}\frac{2(11-2\sqrt6)t_+-(2-3\sqrt6)}{2\sqrt2}}{4\sqrt{2(11-2\sqrt6)}}+C\\
= & \ \frac{\sinh^{-1}\frac{(\sqrt2+3\sqrt3)x -(\sqrt2+\sqrt3)}{2x+\sqrt6}}{4\sqrt{2(11+2\sqrt6)}}+\frac{\sinh^{-1}\frac{(\sqrt2-3\sqrt3)x-(\sqrt2-\sqrt3)}{2x-\sqrt6}}{4\sqrt{2(11-2\sqrt6)}}+C 
\end{align}
