Integration of irrational function

[gruba]

Homework Statement

Find the integral \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx

2. The attempt at a solution
I can't find a useful substitution to solve this integral.
I tried x-2=\frac{1}{u},x=\frac{1}{u}+2,dx=-\frac{1}{u^2}du that gives
\int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx=-\int \frac{u}{\sqrt{3\left(\frac{1}{u}+2\right)^2-8\left(\frac{1}{u}+2\right)+5}}\mathrm du

Is there a useful substitution for u?

 

[SammyS]

Homework Statement

Find the integral \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx

2. The attempt at a solution
I can't find a useful substitution to solve this integral.
I tried x-2=\frac{1}{u},x=\frac{1}{u}+2,dx=-\frac{1}{u^2}du that gives
\int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx=-\int \frac{u}{\sqrt{3\left(\frac{1}{u}+2\right)^2-8\left(\frac{1}{u}+2\right)+5}}\mathrm du

Is there a useful substitution for u?

Factor $\ 3x^2-8x+5\$. Then putting your u substitution into that simplifies things a bit.