MHB Can You Solve This Advanced Definite Integral Using Residue Theorem?

[sbhatnagar]

This is classic one. Prove that

$$ \int_0^\infty \frac{dx}{\left\{x^4+(1+2\sqrt{2})x^2+1 \right\}\left\{x^{100}-x^{99}+x^{98}-\cdots +1\right\}}=\frac{\pi}{2(1+\sqrt{2})}$$

 

[anuttarasammyak]

Residue theorem cound be applicable. The integrand is
[\ (x-\alpha i)(x+\alpha i)(x-\beta i)(x+\beta i)\prod_{j=1}^{100}(x+e^{\pi i (\frac{2j}{101})})\ ]^{-1}
where for $0< \alpha < \beta$ , $-\alpha^2$ and $-\beta^2$ are solutions of quadratic equation,
y^2+(1+2\sqrt{2})y+1=0
But I have not found an appropriate contour.