I cant figure

Help me out, superior math minds?

integral((4x^5 + 19x^4 + 31x^3 - 15x^2 + 85x - 13) / (x^2(x^2 + 6x + 13)))dx

This is from calculus 2, so I cant use hyperbolic trigonometric functions (we haven't used it at all).

It's not as much as I need the answer as much as I need to understand 'how' to solve the integral.

I have worked down the problem with partial fractions and I got this (if it helps any):

2x^2 - 5x + 7ln|x| - ln|x^2| + 2integral(x/(x^2 + 6x + 13))dx + 9ln|x^2+6x+13|

Thank you, in advance for the help.

Assuming you did everything else right,
integral(x/(x^2 + 6x + 13))dx
integral((x+3-3)/(x^2 + 6x + 13))dx
integral((x+3)/(x^2 + 6x + 13))dx - integral(3/((x+3)^2 +4))dx

-> u = x^2 + 6x + 13, du = (2x+6)dx; v = x+3, dv = dx
integral(du/2u) - 3integral(dv/(v^2 + 4))

left thing is obvious, right thing is some arctan nonsense

How did I know to separate the thing?  I did the u substitution first, then found out I had crap left over.

Thank you.  Can't tell you how much I needed the help with that one.