\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {F \left ( \left ( y \left ( x \right ) \right ) ^{3/2}-3/2\,{{\rm e}^{x}} \right ) {{\rm e}^{x}}}{\sqrt {y \left ( x \right ) }}}=0} \]
Mathematica: cpu = 62.104386 (sec), leaf count = 218 \[ \text {Solve}\left [\int _1^{y(x)} \left (\frac {\sqrt {K[2]}}{F\left (K[2]^{3/2}-\frac {3 e^x}{2}\right )-1}-\int _1^x \left (\frac {3 e^{K[1]} \sqrt {K[2]} F\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right ) F'\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )}{2 \left (F\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )-1\right )^2}-\frac {3 e^{K[1]} \sqrt {K[2]} F'\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )}{2 \left (F\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )-1\right )}\right ) \, dK[1]\right ) \, dK[2]+\int _1^x -\frac {e^{K[1]} F\left (y(x)^{3/2}-\frac {3 e^{K[1]}}{2}\right )}{F\left (y(x)^{3/2}-\frac {3 e^{K[1]}}{2}\right )-1} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.265 (sec), leaf count = 35 \[ \left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\!{1\sqrt {{\it \_a}} \left ( F \left ( {{\it \_a}}^{{\frac {3}{2}}}-{\frac {3\,{{\rm e}^{x}} }{2}} \right ) -1 \right ) ^{-1}}\,{\rm d}{\it \_a}-{{\rm e}^{x}}-{\it \_C1}=0 \right \} \]