Integrate:

$\displaystyle \int \frac{2x^5 - 162x + 3}{x^4 - 81}dx$

Now I used partial fractions:

$\displaystyle \frac{2x^5 - 162x + 3}{x^4 - 81} = \frac{A}{x - 3} + \frac{B}{x + 3} + \frac{Dx + E}{x^2 + 9}$

and after a lot of working out got the following equations:

$\displaystyle A + B + D = 0$

$\displaystyle 3A - 3B + E = 12$

$\displaystyle 9A + 9B - 9D = -162$

$\displaystyle

27A - 27B - 9E = 3$

Now I tried gauss reducing it, but it was rather long and tedious which gave me:

$\displaystyle A = \frac{-167}{36}$

$\displaystyle B = \frac{157}{-36}$

$\displaystyle D = 9$

$\displaystyle E = -\frac{1}{6}$

In my final answer I got:

$\displaystyle \frac{-167}{36}ln|x - 3| - \frac{157}{36}ln|x + 3| + \frac{9}{2}ln|x^2 + 9| - \frac{1}{54}arctan(\frac{x}{3}) + C$

which was of that right form (due to the logs and arctan) but obviously the values were wrong.

So I was wondering, is there a more efficient way to do this question and solve for A, B, D and E without using gauss reduction?