$\displaystyle \int \frac{dx}{x^4\sqrt{1+x^2}}$
answer:
$\displaystyle \frac{(2x^2-1)(1+x^2)^\frac{1}{2}}{3x^3}$
There are many ways, but I like trig sub for some reason.
Let $\displaystyle x=tan(u), \;\ dx=sec^{2}(u)du$
When we make the subs we get:
$\displaystyle \int csc(u)cot^{3}(u)du$
$\displaystyle =\int csc(u)(csc^{2}(u)-1)cot(u)du$
Now, can you continue?. Another sub should do it, and when you get to the end remember to let $\displaystyle u=tan^{-1}(x)$ to get it back in terms of x.