Evalulate $\int_0^\infty \int_0^\infty \int_0^\infty e^{-(xy+yz+zx)}\ dx\ dy\ dz$

Here's another approach: By making substitution $$ (u,v,w) = (xy,yz,zx), $$ we have$$\mathrm du \mathrm dv\mathrm dw = 2xyz \mathrm dx\mathrm dy\mathrm dz\implies \frac{\mathrm du \mathrm dv\mathrm dw}{2\sqrt{uvw}}=\mathrm dx\mathrm dy\mathrm dz.$$ Hence, $$\begin{align*} \int_0^\infty \int_0^\infty \int_0^\infty e^{-(xy+yz+zx)}\mathrm dx\mathrm dy\mathrm dz&=\frac12\int_0^\infty \int_0^\infty \int_0^\infty \frac{e^{-u-v-w}\mathrm du \mathrm dv\mathrm dw}{\sqrt{uvw}}\\&=\frac12\left( \int_0^\infty \frac{e^{-u}\mathrm du}{\sqrt{u}}\right)^3\\&=\frac12\left(2 \int_0^\infty e^{-v^2}\mathrm dv\right)^3\\ &=\frac{\pi^{\frac32}}{2}. \end{align*}$$

A chain of mostly elementary manipulations: \begin{align*}I &= \int_0^{\infty}\int_0^{\infty}\int_0^{\infty}e^{-(xy+yz+zx)}\,dx\,dy\,dz\\ &= \int_0^{\infty}\int_0^{\infty}\int_0^{\infty}e^{-yz}e^{-x(y+z)}\,dx\,dy\,dz\\ &= \int_0^{\infty}\int_0^{\infty}\frac{e^{-yz}}{y+z}\,dy\,dz\end{align*} First, we evaluate the inner integral with elementary techniques. There's some clutter that amounts to constant multipliers, but it's just $e^{ax}$ at its core. Next, to handle $y$, we substitute $y=zu$, $dy=z\,du$: \begin{align*}I &= \int_0^{\infty}\int_0^{\infty} \frac{e^{-z^2u}}{z(1+u)}\cdot z\,du\,dz\\ &= \int_0^{\infty}\int_0^{\infty} \frac{e^{-uz^2}}{1+u}\,dz\,du\\ &= \int_0^{\infty}\frac{\frac{\sqrt{\pi}}{2\sqrt{u}}}{1+u}\,du\end{align*} After the substitution, the $u$ integral isn't something we want to deal with - so we switch the order and do the $z$ integral next. That's just a Gaussian - no elementary antiderivative, but we know that $\int_0^{\infty} e^{-az^2}\,dz=\frac{\sqrt{\pi}}{2\sqrt{a}}$. This leaves us with a $u$ integral that's algebraic; we substitute $v=\sqrt{u}$, $dv=\frac1{2\sqrt{u}}\,du$ to rationalize it: $$I = \sqrt{\pi}\int_0^{\infty}\frac1{1+v^2}\,dv = \sqrt{\pi}\cdot\frac{\pi}{2}=\frac{\pi^{\frac32}}{2}$$ Done.

There's probably a way to connect this to a three-dimensional Gaussian, but I found it easier to break it down this way.