Provided that $(\alpha, \beta) \in \mathbb{R}_+^2$, is the following integral
$$ \int_{0}^{+\infty}{x^{\beta-\alpha-1} \exp\left(-\beta x - \frac{\alpha}{x} \right) dx} $$
convergent?
I know that the integrand is continuous on $]0,+\infty[$ and is equivalent to $x^{\beta-\alpha-1} \exp(-\beta x)$ near $x=+\infty$ and $x^{\beta-\alpha-1} \exp(-\frac{\alpha}{x})$ near $x=0$. So this should be enough to confirm the convergence?