\[ y'(x)=\frac {x^2+2 x y(x)+e^{2 (x-y(x))^2 (y(x)+x)^2}+y(x)^2}{x^2+2 x y(x)-e^{2 (x-y(x))^2 (y(x)+x)^2}+y(x)^2} \] ✓ Mathematica : cpu = 1.89132 (sec), leaf count = 228
DSolve[Derivative[1][y][x] == (E^(2*(x - y[x])^2*(x + y[x])^2) + x^2 + 2*x*y[x] + y[x]^2)/(-E^(2*(x - y[x])^2*(x + y[x])^2) + x^2 + 2*x*y[x] + y[x]^2),y[x],x]
\[\text {Solve}\left [\int _1^{y(x)}\left (-\frac {2 K[2]}{-x^2+e^{2 (x-K[2])^2 (x+K[2])^2}+K[2]^2}-\int _1^x\left (\frac {2 K[1] \left (-2 K[2]-e^{2 (K[1]-K[2])^2 (K[1]+K[2])^2} \left (4 (K[1]-K[2])^2 (K[1]+K[2])-4 (K[1]-K[2]) (K[1]+K[2])^2\right )\right )}{\left (K[1]^2-e^{2 (K[1]-K[2])^2 (K[1]+K[2])^2}-K[2]^2\right )^2}-\frac {1}{(K[1]+K[2])^2}\right )dK[1]+\frac {1}{x+K[2]}\right )dK[2]+\int _1^x\left (\frac {1}{K[1]+y(x)}-\frac {2 K[1]}{K[1]^2-e^{2 (K[1]-y(x))^2 (K[1]+y(x))^2}-y(x)^2}\right )dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.287 (sec), leaf count = 38
dsolve(diff(y(x),x) = (y(x)^2+2*x*y(x)+x^2+exp(2*(x-y(x))^2*(y(x)+x)^2))/(y(x)^2+2*x*y(x)+x^2-exp(2*(x-y(x))^2*(y(x)+x)^2)),y(x))
\[y \left (x \right ) = {\mathrm e}^{\RootOf \left (-\textit {\_Z} +\int _{}^{{\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} x}\frac {1}{{\mathrm e}^{2 \textit {\_a}^{2}}+\textit {\_a}}d \textit {\_a} +c_{1}\right )}-x\]