Evaluate $\int\limits_{}^{}\frac{x (1-ax^2)^N}{(1/x^{\gamma}-t)^2}dx$

Question

Evaluate $$I(\eta)=\int\limits_{x=\eta}^{2}\frac{x (1-ax^2)^N}{(1/x^{\gamma}-t)^2}dx$$ where $a>0,\gamma>2,t>0,N\in{\mathbb{N}}, \eta>0$.

Any idea how to attack this integral? This comes from an expected value $\mathbb{E}\left[\frac{1}{(1/x^{\gamma}-t)^2}\right]$.