for $b=0$ one has
$$\int x^{-a} \exp \left[- \frac{( cx^{-d})^2}{2} \right] dx=\frac{1}{2 d}x^{1-a} E_{1-\frac{a-1}{2 d}}\left(\tfrac{1}{2} c^2 x^{-2 d}\right),$$
with $E_n(x)$ the exponential integral function.