Express the integral

$$I=\int_{0}^{\infty} d x \int_{0}^{1} d y \int_{0}^{x} d z x e^{-A x / y-B x y-C y z}$$

in terms of the new variables $\alpha=x / y, \beta=x y$, and $\gamma=y z$. Hence show that

$$I=\frac{1}{2 A(A+B)(A+B+C)}$$

You may assume $A, B$ and $C$ are positive. [Hint: Remember to calculate the limits of the integral.]