= Solution
Translate the variables by $u=x-x_0$, $v=y-y_0$, $w=z-z_0$. The numerator becomes $u+x_0$. The part odd in $u$ integrates to zero, while the constant part factors into three <Gaussian integrals>. Thus
$$
\begin{aligned}
I&=x_0\left(\int_{-\infty}^{\infty}e^{-u^2/a^2}du\right)^3\\
&=x_0(a\sqrt\pi)^3
=\boxed{\pi^{3/2}a^3x_0.}
\end{aligned}
$$
The absolute integrability of a <polynomial> times a Gaussian justifies this separation by <Fubini's theorem>. Neither $y_0$ nor $z_0$ affects the result; when $x_0=0$, odd symmetry makes the integral zero.
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