$\displaystyle \begin{align*} \left( \frac{3}{2}x^2 - 6x^{-2} \right) ^2 = \frac{9}{4}x^4 - 18 + 36x^{-4} \end{align*}$, not $\displaystyle \begin{align*} \frac{9}{4}x^4 - 18x + 36x^{-4} \end{align*}$.

Once you have fixed this up in your integrand, get a common denominator and simplify.