Question

-3x^3+8x^2+24x-39 /(3x-2)^2 x (x^2-2x+5) integrate by partial fraction decomposition

Answer 1

What a cumbersome and ridiculous question!

First of all, due to your lack of brackets, I will assume you meant
(-3x^3+8x^2+24x-39) / ((3x-2)^2 * (x^2-2x+5) )

So you need this:
(-3x^3+8x^2+24x-39) / ((3x-2)^2 * (x^2-2x+5) )
= A/(3x-2) + (Bx+C)/(3x-2)^2 + (Dx+E)/(x^2 - 2x + 5)

Multiply each term by (3x-2)^2 (x^2 - 2x + 5)
then create 5 equations in 5 variables, using
x = 2/3, x=0, x=1, x=-1, x=2

I did this on paper but I am not going to type all that out.
Here is what Wolfram got and it agrees with mine:

www.wolframalpha.com/input?i=partial+fraction+%28-3x%5E3%2B8x%5E2%2B24x-39+%29%2F%28%283x-2%29%5E2+%28x%5E2-2x%2B5%29%29

Now enjoy integrating those 3 terms.

Here is what Wolfram said about that:

www.wolframalpha.com/input?i=%E2%88%AB+%28-3x%5E3%2B8x%5E2%2B24x-39+%29%2F%28%283x-2%29%5E2+%28x%5E2-2x%2B5%29%29+dx

Answer 2

I'm assuming you found C=0
You only need a linear numerator when the denominator is an irreducible quadratic.

Answer 3

actually, I meant B=0
:-(