I am interested in find the probability density function corresponding to the characteristic function

$\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$

where $c > 1$ and and $0< b < 1$. Clearly when $b=0$ this is just the Gamma distribution with shape parameter $c$ and scale parameter $1$, and for $b \neq 0$, the solution can be found from simple contour integration if $c$ is an integer. The case I am unsure of is when $b>0$ and $c \notin \mathbb{Z}$, and thus I need to compute

$p(y) = \frac{1}{2 \pi} \int_{-\infty}^\infty \left(\frac{1 - i b t}{1 - i t}\right)^c e^{-iyt} \, dt, \, \, \,\,\,\, c\notin \mathbb{Z}, \, c > 1, \, 0<b<1$

Since this function will be multi-valued, I was thinking of proceeding by contour integration using a keyhole contour similar to the one used in this or this problem, but the integral on the small semi-circular contour around the branch point $C_\epsilon$ does not vanish in this case, because the function essentially has a pole of non-integer power there. Any help in this would be appreciated. Moreover, if someone simply knows this result and can refer me to a table of integrals which has it, that would be sufficient.

-Ben