How to compute $\int{\frac{5x^3+8x^2+x+2}{x^2(2x^2+1)}} dx$? $$\int{\frac{5x^3+8x^2+x+2}{x^2(2x^2+1)}} dx$$
So ... how do I start? Numerator cant be factorized it seems, and this looks like a complicated expression ... 
I tried expanding the denominator to see if integration by substitution will work, but it didn't give any ideas
$$\int{\frac{5x^3+8x^2+x+2}{2x^4+x^2}} dx$$
I don't see a clear way to integrate by parts either. Can anyone enlighten me?

If you wish to avoid partial fractions, it is possible to clean up that denominator with a simple substitution.  To make that $2x^2+1$ factor more manageable, make the substitution
$x=\frac{\sqrt2}2\tan\theta,dx=\frac{\sqrt2}2\sec^2\theta d\theta$
$\int\dfrac{5x^3+8x^2+x+2}{x^2(2x^2+1)}dx=\int\dfrac{\frac{5\sqrt2}4\tan^3\theta+4\tan^2\theta+\frac{\sqrt2}2\tan\theta+2}{\frac12\tan^2\theta(\tan^2\theta+1)}\times\frac{\sqrt2}2\sec^2\theta d\theta$
$\dfrac{\frac{\sqrt2}2\sec^2\theta}{\frac12(\tan^2+1)}$ reduces to $\sqrt2$.  Then dividing through gets
$\int\frac52\tan\theta+4\sqrt2+\cot\theta+2\sqrt2\cot^2\theta d\theta$
Can you take it from there?
