consider (c), which is the most general, and consider a "pizza contour" $\gamma$ with an angle $\theta$ such that $e^{i\beta\theta}=1$ (it may be greater than $2\pi$, but it really does not matter) and also if $-1<\gamma <0$ you should cut a small piece of the radius $\varepsilon$. Consider

$$

\int_\gamma \frac{z^\gamma\,dz}{1+z^\beta}.

$$

What are singularities inside? Calculate the residue (or residues).

Prove that for $R\to \infty$ the integral over big arc tends to $0$

Prove that for $\varepsilon\to 0$ the integral over big arc tends to $0$

Express integrals over straight segments via

$$\int_\varepsilon ^R \frac{x^\gamma\,dx}{1+x^\beta}.$$

Then after taking the limits you'll be able to find integral in question.

There are Examples in the section, using exactly the same method.