TITLE: Help solving $\int \:\frac{dx}{x+\sqrt{9x^2-9x+2}}$
QUESTION [11 upvotes]: Reading an article online I came across this integral I've been trying to solve since yesterday morning and I literally tried every integration method I know and I still can't solve it. I am a beginner to integrals so help will be appreciated!
I tried substitution, integration by parts and I can't seem to find the correct way!
$$\int \:\frac{dx}{x+\sqrt{9x^2-9x+2}}$$
Does anyone know how to proceed from here? What to do? Because I am clueless. Thanks!

REPLY [7 votes]: The Euler substitution $\sqrt{9x^2-9x+2} =t+3x$ is better suited here, which results in $x=\frac{2-t^2}{3(2t+3)} $, $dx=-\frac23 \frac{t^2+3t+2}{(2t+3)^2}dt$ and
\begin{align}
&\int \:\frac{1}{x+\sqrt{9x^2-9x+2}}dx\\
=& -2 \int \frac{t^2+3t+2}{(2t+3)(2t^2+9t+8)}dt\\
=&-\frac12 \int\left(\frac1{2t+3}+\frac t{2t^2+9t+8} \right)dt\\
=& -\frac14\ln|2t+3| -\frac1{8}\ln| 2t^2+9t+8|+\frac9 {8\sqrt{17}}\ln|\frac{4t+9-\sqrt{17}}{4t+9+\sqrt{17}} |+C
\end{align}