\[ x^{\frac {n}{n+1}} y''(x)-y(x)^{\frac {2 n+1}{n+1}}=0 \] ✗ Mathematica : cpu = 0.0664395 (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 = 4.227 (sec), leaf count = 156
\[\left \{y \left (x \right ) = \mathit {ODESolStruc} \left (\textit {\_a} \,{\mathrm e}^{-\frac {\left (c_{1}+\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} \right ) \left (n +2\right )}{n}}, \left [\left \{\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )=\frac {2 \left (-\frac {n^{2} \textit {\_a}^{\frac {2 n +1}{n +1}} \textit {\_}b\left (\textit {\_a} \right )}{2}+\left (n +2\right ) \left (n +1\right ) \textit {\_a} \textit {\_}b\left (\textit {\_a} \right )-\frac {3 n^{2}}{2}-2 n \right ) \textit {\_}b\left (\textit {\_a} \right )^{2}}{n^{2}}\right \}, \left \{\textit {\_a} =x^{\frac {n +2}{n}} y \left (x \right ), \textit {\_}b\left (\textit {\_a} \right )=\frac {n \,x^{-\frac {n +2}{n}}}{n x \left (\frac {d}{d x}y \left (x \right )\right )+\left (n +2\right ) y \left (x \right )}\right \}, \left \{x ={\mathrm e}^{c_{1}+\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a}}, y \left (x \right )=\textit {\_a} \,{\mathrm e}^{-\frac {\left (c_{1}+\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} \right ) \left (n +2\right )}{n}}\right \}\right ]\right )\right \}\]