\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +a{{\rm e}^{x}}\sqrt {y \left ( x \right ) }=0} \]
Mathematica: cpu = 0.519566 (sec), leaf count = 22 \[ \text {DSolve}\left [a e^x \sqrt {y(x)}+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.811 (sec), leaf count = 109 \[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\frac {{\it \_a }}{{{\rm e}^{-2\,\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{ \it \_a}-2\,{\it \_C1}}}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}} {\it \_b} \left ( {\it \_a} \right ) = \left ( \sqrt {{\it \_a}}a+4\,{ \it \_a} \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^ {3}+4\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {{\rm e}^{-2\,x}},{ \it \_b} \left ( {\it \_a} \right ) =-{\frac {1}{{{\rm e}^{-2\,x}} \left ( 2\,y \left ( x \right ) -{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) }} \right \} , \left \{ x=\int \!{\it \_b} \left ( { \it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1},y \left ( x \right ) ={ \frac {{\it \_a}}{{{\rm e}^{-2\,\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}-2\,{\it \_C1}}}}} \right \} ] \right ) \right \} \]