= Solution
Set $x=au$, $y=bv$, $z=cw$. For nonzero $a,b,c$, the absolute <Jacobian determinant> is $|abc|$, and the transformed interior is the ball $u^2+v^2+w^2\le4$. Its radius is two, so
$$
V=|abc|\int_0^2\int_0^\pi\int_0^{2\pi}\rho^2\sin\theta\,d\phi\,d\theta\,d\rho
=\boxed{\frac{32\pi}{3}|abc|.}
$$
If $a,b,c$ are taken as positive semiaxis parameters, this is $32\pi abc/3$. The actual semiaxes are $2|a|,2|b|,2|c|$; omitting the factor two in the transformed radius would make the volume eight times too small.
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