How can I use complex analysis to compute the integral of $\int_0^\infty \frac{x^\alpha}{1 + (kx)^\beta} dx$

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How can I use complex analysis to compute the integral of

$$\int_0^\infty \frac{x^\alpha}{1 + (kx)^\beta} dx$$

I would like to see how to do it using keyhole integration or some other method.