Alternative way:

 $Version
 (* "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)" *)

 a = 2; b = 1.5; c = 1; tx = 0.175; ty = 0.2; tz = 0.25;
 R = ImplicitRegion[Exp[tx*x] (x/a)^2 + Exp[ty*y] (y/b)^2 + Exp[tz*z] (z/c)^2 <= 1, {x,y, z}];
 V = DiscretizeRegion[R, Method -> "RegionPlot3D"] // Quiet;
 Volume[V]

(* 6.924361674 *)

or:

a = 2; b = 1.5; c = 1; tx = 0.175; ty = 0.2; tz = 0.25;
AbsoluteTiming[N[Volume[ImplicitRegion[Exp[tx*x]*(x/a)^2 + Exp[ty*y]*(y/b)^2 + Exp[tz*z]*(z/c)^2 <= 1, {x, y, z}]]]]

(*  {202.8073994, 6.924361674} *)

I had to correct the answer because I had a small mistake.

Regards,MI