compound of gamma and exponential distribution

If $X$ and $Y \mid X$ are parametrized in terms of rate, then the marginal distribution of $Y$ can be computed by observing $$\begin{align*} f_Y(y) &= \int_{x=0}^\infty f_{Y \mid X}(y \mid x) f_X(x) \, dx \\ &= \int_{x=0}^\infty x e^{-x y} \frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)} \, dx \\ &= \frac{\beta^\alpha}{\Gamma(\alpha)} \int_{x=0}^\infty x^{\alpha} e^{-(y+\beta)x} \, dx \\ &= \frac{\beta^\alpha \Gamma(\alpha+1)}{\Gamma(\alpha)(y+\beta)^{\alpha+1}} \int_{x=0}^\infty \frac{(y+\beta)^{\alpha+1} x^{\alpha} e^{-(y+\beta)x}}{\Gamma(\alpha+1)} \, dx \\ &= \frac{\alpha\beta^\alpha}{(y + \beta)^{\alpha+1}} . \end{align*}$$ This is a (shifted) Pareto distribution on $Y \in [0, \infty)$.