The first step is to make all polynomial fractions into proper fractions, meaning that the degree of the denominator must exceed that of the numerator. But this you already have.
The next step is to factorise the polynomial in the denominator as much as possible. In this case a possible factorisation is
If we stick with real numbers, the last factor has no real roots (check this), so we keep it as it is.
Next we do partial fractions, and make the ansatz
(see the link below for why the second factor has
Multiply both sides by the denominator in the left hand side to get
Now for this to hold for any
Then we get a system of equations
Solving for the constants, we get that
Thus we have shown that
The right hand side we can integrate using standard integrals. The indeterminate integral (unique up to a constant) is then
Note 1: http://www.purplemath.com/modules/partfrac2.htm has a lot of good examples and presents some alternative methods.