Fermi-dirac integral with $1/p$ dependence in the exponent: $1/(1+\exp(a\cdot p+b/p+c))$

Question

I am having some trouble solving the following integral.

$$\int_{0}^{\infty}\frac{dp}{1+e^{a*p+b/p+c}}$$

This should be solve for two situations: $b > 0$ and $b < 0$.

In the derivation it may be used that $ap >> b/p$, which is the limitation for the validity of the additional term encompassing 1/p.

Can anyone help?

Thanks in advance.

Edit: The limitation $a*p >> b/p$ is very subtle.