contour integral, limiting contour theorem with residue

Hi, can anyone please help me with this question, thanks a lot.

$\displaystyle \displaystyle \int_{0}^{\infty} \frac{x^{-1/3}}{x^2+1} dx$

I did try to take the contour, and take notice the three "bad points" are $\displaystyle 0$, $\displaystyle i$, and $\displaystyle -i$.

I used residue theorem that $\displaystyle \displaystyle\oint_{\Gamma_{R,\epsilon}} \frac{dz}{\sqrt[3]{z}(z^2+1)}=2\pi i\displaystyle \sum_{poles\ in\ the\ plane}Res(f(z), a_j)$.

I can use limiting contour theorem to get one integral is $\displaystyle 0$.

However, I'm really having trouble solve this question, I thought my methods are right, but I can't get the right answer which is $\displaystyle \frac{\sqrt{3}{\pi}}{3}$. One friend told me I need to worry about choosing branch because of that $\displaystyle \sqrt[3]{z}$, but I don't quite understand it and what I supposed to do.

Can anyone please show me some precise steps on solving this question? Thanks a lot.