\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( y \left ( x \right ) +3\,a \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{3}+a \left ( y \left ( x \right ) \right ) ^{2}+2\,{a}^{2}y \left ( x \right ) =0} \]
Mathematica: cpu = 26.601378 (sec), leaf count = 41 \[ \text {DSolve}\left [2 a^2 y(x)+(3 a+y(x)) y'(x)+a y(x)^2+y''(x)-y(x)^3=0,y(x),x\right ] \]
Maple: cpu = 0.219 (sec), leaf count = 775 \[ \left \{ y \left ( x \right ) ={\frac {1}{{{\rm e}^{ax}}}{\it RootOf} \left ( \int ^{{\it \_Z}}\!{\frac {1}{-{{\it \_f}}^{6}+{\it \_C1}} \left ( {{\it \_f}}^{8}-{\it \_C1}\,{{\it \_f}}^{2}+ \left ( -{{\it \_f }}^{12}+2\,{\it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{12}-2\,\sqrt { {\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{{\it \_f}}^{6}+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{ \it \_C1}}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{ -{{\it \_f}}^{12}+2\,{\it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+ \sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{ 12}-2\,\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{{\it \_f}}^{6}+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{ \it \_C1}}}}{{\it \_C1}}^{2}}}}}{d{\it \_f}}a+{\it \_C2}\,a+{{\rm e}^{ -ax}} \right ) },y \left ( x \right ) ={\frac {1}{{{\rm e}^{ax}}}{\it RootOf} \left ( \int ^{{\it \_Z}}\!{\frac {1}{-{{\it \_f}}^{6}+{\it \_C1}} \left ( -i\sqrt {3}{{\it \_f}}^{8}-{{\it \_f}}^{8}+i\sqrt {3}{ \it \_C1}\,{{\it \_f}}^{2}+i\sqrt {3} \left ( -{{\it \_f}}^{12}+2\,{ \it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+\sqrt {{\frac {{\it \_C1} }{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{12}-2\,\sqrt {{\frac {{ \it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{{\it \_f}}^{6}+ \sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_C1}}^ {2} \right ) ^{{\frac {2}{3}}}+{\it \_C1}\,{{\it \_f}}^{2}- \left ( -{{ \it \_f}}^{12}+2\,{\it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+\sqrt { {\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{12}-2\, \sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{ {\it \_f}}^{6}+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}} }}{{\it \_C1}}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{-{{\it \_f}}^{12}+2\,{\it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+ \sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{ 12}-2\,\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{{\it \_f}}^{6}+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{ \it \_C1}}}}{{\it \_C1}}^{2}}}}}{d{\it \_f}}a+2\,{\it \_C2}\,a+2\,{ {\rm e}^{-ax}} \right ) },y \left ( x \right ) ={\frac {1}{{{\rm e}^{ax}} }{\it RootOf} \left ( -\int ^{{\it \_Z}}\!{\frac {1}{-{{\it \_f}}^{6}+{ \it \_C1}} \left ( -i\sqrt {3}{{\it \_f}}^{8}+{{\it \_f}}^{8}+i\sqrt {3 }{\it \_C1}\,{{\it \_f}}^{2}+i\sqrt {3} \left ( -{{\it \_f}}^{12}+2\,{ \it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+\sqrt {{\frac {{\it \_C1} }{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{12}-2\,\sqrt {{\frac {{ \it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{{\it \_f}}^{6}+ \sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_C1}}^ {2} \right ) ^{{\frac {2}{3}}}-{\it \_C1}\,{{\it \_f}}^{2}+ \left ( -{{ \it \_f}}^{12}+2\,{\it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+\sqrt { {\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{12}-2\, \sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{ {\it \_f}}^{6}+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}} }}{{\it \_C1}}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{-{{\it \_f}}^{12}+2\,{\it \_C1}\,{{\it \_f}}^{6}-{{\it \_C1}}^{2}+ \sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{{\it \_f}}^{ 12}-2\,\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}}{\it \_C1}\,{{\it \_f}}^{6}+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{ \it \_C1}}}}{{\it \_C1}}^{2}}}}}{d{\it \_f}}a+2\,{\it \_C2}\,a+2\,{ {\rm e}^{-ax}} \right ) } \right \} \]