$\int\frac{(a-i\omega)\cdot(b-i\omega)\cdot e^{iwt}}{(a-i\omega)\cdot(b-i\omega)-(1-V^{-c-\sqrt{d -2\omega}})\cdot(1-V^{-c-\sqrt{e-2\omega}})}d\omega$

Question

I am having trouble with one integral, which I don't even know if its possible to calculate:

$$ \int_{-\infty}^\infty \frac{(a - i\cdot\omega)\cdot (b - i\cdot\omega) \cdot e^{iwt}}{(a - i\cdot\omega)\cdot (b - i\cdot\omega) -(1 - V^{-c-\sqrt{d -2\omega}}) \cdot (1 - V^{-c-\sqrt{e -2\omega}}) } \, d\omega ,$$

where $a,b,c,d,e, V >0$. This is inverse fourier transform that came from solving a system of integral equations. I have tried several things in Mathematica but I haven't been able to solve it. Do you know any tips that might help?