\[ \boxed { {x}^{{\frac {n}{n+1}}}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{{\frac {2\,n+1}{n+1}}}=0} \]
Mathematica: cpu = 0.083011 (sec), leaf count = 37 \[ \text {DSolve}\left [x^{\frac {n}{n+1}} y''(x)-y(x)^{\frac {2 n+1}{n+1}}=0,y(x),x\right ] \]
Maple: cpu = 3.370 (sec), leaf count = 165 \[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {{\it \_a} \left ( {{\rm e}^{{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( n+2 \right ) }{n }}}} \right ) ^{-1}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =-{\frac { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3}}{{n}^{2}} \left ( {{\it \_a}}^{{\frac {2\,n+ 1}{n+1}}}{n}^{2}-2\,{\it \_a}\,{n}^{2}-6\,{\it \_a}\,n-4\,{\it \_a} \right ) }-{\frac { \left ( 3\,n+4 \right ) \left ( {\it \_b} \left ( { \it \_a} \right ) \right ) ^{2}}{n}} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{{\frac {n+2}{n}}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {n}{nx{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +ny \left ( x \right ) +2\,y \left ( x \right ) } \left ( {x}^{{\frac {n+2}{n} }} \right ) ^{-1}} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={{\it \_a} \left ( {{\rm e}^{{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( n+2 \right ) }{n}}}} \right ) ^{-1}} \right \} ] \right ) \right \} \]