\[ x^{\frac {n}{n+1}} y''(x)-y(x)^{\frac {2 n+1}{n+1}}=0 \] ✗ Mathematica : cpu = 0.0540371 (sec), leaf count = 0 , could not solve
DSolve[-y[x]^((1 + 2*n)/(1 + n)) + x^(n/(1 + n))*Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 17.604 (sec), leaf count = 156
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {{\it \_a} \left ( {{\rm e}^{{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( n+2 \right ) }{n}}}} \right ) ^{-1}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =2\,{\frac { \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}}{{n}^{2}} \left ( -1/2\,{{\it \_a}}^{{\frac {2\,n+1}{n+1}}}{\it \_b} \left ( {\it \_a} \right ) {n}^{2}+{\it \_a}\, \left ( n+2 \right ) \left ( n+1 \right ) {\it \_b} \left ( {\it \_a} \right ) -3/2\,{n}^{2}-2\,n \right ) } \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{{\frac {n+2}{n}}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {n}{nx{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +y \left ( x \right ) \left ( n+2 \right ) } \left ( {x}^{{\frac {n+2}{n}}} \right ) ^{-1}} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={{\it \_a} \left ( {{\rm e}^{{\frac { \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) \left ( n+2 \right ) }{n}}}} \right ) ^{-1}} \right \} ] \right ) \right \} \]