\[ \left (y'(x)^2+y(x)^2\right ) y''(x)+y(x)^3=0 \] ✓ Mathematica : cpu = 0.548676 (sec), leaf count = 371
DSolve[y[x]^3 + (y[x]^2 + Derivative[1][y][x]^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_2 \exp \left (\frac {1}{12} \left (-2 \sqrt {3} \arctan \left (\frac {1+2 \text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ][-x+c_1]{}^2}{\sqrt {3}}\right )-3 \log \left (\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ][-x+c_1]{}^4+\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ][-x+c_1]{}^2+1\right )\right )\right )\right \}\right \}\] ✓ Maple : cpu = 1.597 (sec), leaf count = 159
dsolve((diff(y(x),x)^2+y(x)^2)*diff(diff(y(x),x),x)+y(x)^3=0,y(x))
\[y \left (x \right ) = \frac {\sqrt {c_{1}+\tan \left (\sqrt {3}\, x \right )}\, {\mathrm e}^{\frac {\sqrt {3}\, \left (\int \frac {\sqrt {\left (9 c_{1}^{2}+12\right ) \sec \left (\sqrt {3}\, x \right )^{2}+3 c_{1}^{2}+6 c_{1} \tan \left (\sqrt {3}\, x \right )-3}}{c_{1}+\tan \left (\sqrt {3}\, x \right )}d x \right )}{6}+c_{2}}}{\left (\sec \left (\sqrt {3}\, x \right )^{2}\right )^{\frac {1}{4}}}\]