Can anybody help me with this integral? I need an analytical solution for this, and maybe $\textit{Mathematica}$ can help too:
\begin{equation} \int\limits_{x=\sqrt{\alpha}}^{\infty} e^{-(ax^2+\frac{b}{x^2})} ~\sqrt{\frac{x^4+cx^2+d}{x^4-\alpha x^2}} ~~dx. \end{equation}
where $a$, $b$, $c$, $d$, $\alpha$ are positive real numbers.