\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {y \left ( x \right ) \left ( \left ( y \left ( x \right ) \right ) ^{2}+{{\rm e}^{-{x}^{2}}}y \left ( x \right ) + \left ( {{\rm e}^{-{x}^{2}}} \right ) ^{2} \right ) x}{ \left ( {{\rm e}^{-{x}^{2}}} \right ) ^{2}}}=0} \]
Mathematica: cpu = 0.171522 (sec), leaf count = 139 \[ \text {Solve}\left [-\frac {25}{3} \text {RootSum}\left [-25 \text {$\#$1}^3+24 \sqrt [3]{-1} 5^{2/3} \text {$\#$1}-25\& ,\frac {\log \left (\frac {3 e^{2 x^2} x y(x)+e^{x^2} x}{5^{2/3} \sqrt [3]{-e^{3 x^2} x^3}}-\text {$\#$1}\right )}{8 \sqrt [3]{-1} 5^{2/3}-25 \text {$\#$1}^2}\& \right ]=c_1-\frac {5 \sqrt [3]{5} e^{x^2} x^3}{18 \sqrt [3]{-e^{3 x^2} x^3}},y(x)\right ] \]
Maple: cpu = 0.141 (sec), leaf count = 122 \[ \left \{ y \left ( x \right ) ={\frac {1}{2\,{{\rm e}^{{x}^{2}}}} \left ( \sqrt {11}\tan \left ( {\it RootOf} \left ( -4\,\sqrt {11}{x}^{2 }+8\,\sqrt {11}\ln \left ( -{\frac {36\,\sqrt {11}}{11}}+36\,\tan \left ( {\it \_Z} \right ) \right ) -4\,\sqrt {11}\ln \left ( {\frac { 2592\,\sqrt {11} \left ( {{\rm e}^{{x}^{2}}} \right ) ^{2}\tan \left ( { \it \_Z} \right ) }{25}}-{\frac {2592\,{{\rm e}^{2\,{x}^{2}}}\sqrt {11} \tan \left ( {\it \_Z} \right ) }{25}}+{\frac {14256\,{{\rm e}^{2\,{x}^{ 2}}} \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}}{25}}+{\frac { 2592\, \left ( {{\rm e}^{{x}^{2}}} \right ) ^{2}}{5}}+{\frac {1296\,{ {\rm e}^{2\,{x}^{2}}}}{25}} \right ) +4\,\sqrt {11}\ln \left ( 11 \right ) +9\,\sqrt {11}{\it \_C1}-8\,{\it \_Z} \right ) \right ) -1 \right ) } \right \} \]