\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -{\frac { \left ( 3\,n+4 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{n}}-2\,{\frac { \left ( n+1 \right ) \left ( n+2 \right ) y \left ( x \right ) }{{n}^{2}} \left ( \left ( y \left ( x \right ) \right ) ^{{\frac {n}{n+1}}}-1 \right ) }=0} \]
Mathematica: cpu = 118.097997 (sec), leaf count = 50 \[ \text {DSolve}\left [-\frac {2 (n+1) (n+2) y(x) \left (y(x)^{\frac {n}{n+1}}-1\right )}{n^2}-\frac {(3 n+4) y'(x)}{n}+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 3.526 (sec), leaf count = 116 \[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( { \it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) -{\frac { 1}{{n}^{2}} \left ( 2\,{{\it \_a}}^{{\frac {n}{n+1}}}{\it \_a}\,{n}^{2} +3\,{\it \_b} \left ( {\it \_a} \right ) {n}^{2}+6\,{{\it \_a}}^{{\frac {n}{n+1}}}{\it \_a}\,n-2\,{\it \_a}\,{n}^{2}+4\,{\it \_b} \left ( {\it \_a} \right ) n+4\,{{\it \_a}}^{{\frac {n}{n+1}}}{\it \_a}-6\,{\it \_a} \,n-4\,{\it \_a} \right ) }=0 \right \} , \left \{ {\it \_a}=y \left ( x \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}x} }y \left ( x \right ) \right \} , \left \{ x=\int \! \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C1}, y \left ( x \right ) ={\it \_a} \right \} ] \right ) \right \} \]