fxy = Exp[-1*(((x - ux)/sx)^2 + ((y - uy)/sy)^2 - 
          2 r ((x - ux)*(y - uy)/(sx*sy)))/(2 (1 - r^2))]/(2*Pi*sx*
       sy*(1 - r^2)^0.5);
  fy = (1/(sy (2*Pi)^0.5))*Exp[-(1/2)*((y - uy)/sy)^2];
  f = Rationalize[x^2*y*(fxy/fy), 0] // Simplify // PowerExpand

  (*(35567099 E^((sy (ux - x) + r sx (-uy + y))^2/(
   2 (-1 + r^2) sx^2 sy^2)) x^2 y)/(89153496 Sqrt[1 - r^2] sx)*)

 INT1 = Integrate[f, {y, -Infinity, Infinity}, 
   Assumptions -> {ux > 0, uy > 0, sx > 0, sy > 0, 0 < r < 1, 
     x \[Element] Reals}]

 (*(35567099 Sqrt[\[Pi]/2] sy x^2 (r sx uy + 
    sy (-ux + x)))/(44576748 r^2 sx^2)*)

 Integrate[INT1, {x, -Infinity, Infinity}, 
  Assumptions -> {ux > 0, uy > 0, sx > 0, sy > 0, 0 < r < 1}]

 (*"Integral of 1 does not converge on {-\[Infinity],\[Infinity]}"*)

Looks like integral is divergent.

Oh my mistake, but answer is the same as before.

Underscore represents Blank it cannot be used in symbol names. Try removing all underscores

u_x // FullForm
(* Pattern[u, Blank[x]] *)

ux // FullForm
(* ux *)