Consider the following code:
a = 2; b = 1.5; c = 1; tx = 0.175; ty = 0.2; tz = 0.25;
AbsoluteTiming[
Volume[ImplicitRegion[
Exp[tx*x] (x/a)^2 + Exp[ty*y] (y/b)^2 + Exp[tz*z] (z/c)^2 <= 1, {x,
y, z}], Method -> "NIntegrate"]]
{39.1684,
RegionMeasure[
ImplicitRegion[
1/4 E^(0.175 x) x^2 + 0.444444 E^(0.2 y) y^2 + E^(0.25 z) z^2 <=
1, {x, y, z}], 3, WorkingPrecision -> MachinePrecision,
Method -> "NIntegrate"]}