complex analysis - Evaluating the improper integral $ int_{0}^{infty}{frac{x^2}{x^{10} + 1}mathrm dx} $

I am trying to solve the following integral, but I don't have a solution, and the answer I am getting doesn't seem correct.

So I am trying to integrate this:

$$ \int_{0}^{\infty}{\frac{x^2}{x^{10} + 1}\,\mathrm dx} $$

To integrate this, I want to use a contour that looks like a pizza slice, out of a pie of radius R. One edge of this pizza slice is along the positive x-axis, if that makes sense. Since $ z^{10} + 1 $ has 10 zeroes, the slice should only be one tenth of a whole circle. So let's call this contour $ C $. Then:

$$ \int_{C}{\frac{z^2}{z^{10} + 1}\,\mathrm dz} = 2 \pi i\,\operatorname{Res}(\frac{x^2}{x^{10} + 1}, e^{i \pi/10}) $$ This is because this slice contains only one singularity. Furthermore:

$$ \int_{C}{\frac{z^2}{z^{10} + 1}\,\mathrm dz} = \int_0^R{\frac{z^2}{z^{10} + 1}\,\mathrm dz} + \int_\Gamma{\frac{z^2}{z^{10} + 1}\,\mathrm dz} $$

And then, by the M-L Formula, we can say that $ \int_\Gamma{\frac{z^2}{z^{10} + 1}\,\mathrm dz} $ goes to $0$ as $R$ goes to infinity. Evaluating $ 2 \pi i\ \operatorname{Res}(\frac{x^2}{x^{10} + 1}, e^{i \pi/10}) $ I get $ \dfrac{\pi}{e^{i \pi/5}} $. Since this answer isn't real, I don't think this could be correct. What did I do wrong?