It's been years since I've done this, so I hope my help really is help!
Looking at that denominator, though, it seems to me that you might be able to factor out an x, leaving you with x(x^2 + 4). That might then be a good candidate for splitting up into partial fractions, especially since the quadratic factor there is a sum of two squares, which I think gets into certain trigonometric identities.
As a review of partial fractions, to split this rational expression into partial fractions, you solve the following equation for A, B, and C:
(2x^2 - x +4) / (x^3 +4x) = (A / x) + ((Bx + C) / (x^2 + 4))
Note the linear numerator to go with the irreducible quadratic denominator.
Does that help at all?