\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {{{\rm e}^{2/3\,x}}}{y \left ( x \right ) {{\rm e}^{-2/3\,x}}+1}}=0} \]
Mathematica: cpu = 0.191024 (sec), leaf count = 85 \[ \text {Solve}\left [7 \left (-9 c_1+3 \log \left (-\frac {2}{3} e^{-4 x/3} y(x)^2-\frac {2}{3} e^{-2 x/3} y(x)+1\right )+4 x\right )=6 \sqrt {7} \tanh ^{-1}\left (\frac {y(x)+4 e^{2 x/3}}{\sqrt {7} \left (y(x)+e^{2 x/3}\right )}\right ),y(x)\right ] \]
Maple: cpu = 0.686 (sec), leaf count = 52 \[ \left \{ y \left ( x \right ) ={1{\it RootOf} \left ( -{{\rm e}^{{\it RootOf} \left ( -343\, \left ( \tanh \left ( 1/6\, \left ( 4\,{\it \_C1}-4 \,x-3\,{\it \_Z} \right ) \sqrt {7} \right ) \right ) ^{2}+343+98\,{ {\rm e}^{{\it \_Z}}} \right ) }}-3+2\,{\it \_Z}+2\,{{\it \_Z}}^{2} \right ) \left ( {{\rm e}^{-{\frac {2\,x}{3}}}} \right ) ^{-1}} \right \} \]