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.