\[ x^3 y(x)^2 y''(x)+(y(x)+x) \left (x y'(x)-y(x)\right )^3=0 \] ✓ Mathematica : cpu = 33.8098 (sec), leaf count = 248
DSolve[(x + y[x])*(-y[x] + x*Derivative[1][y][x])^3 + x^3*y[x]^2*Derivative[2][y][x] == 0,y[x],x]
\[\text {Solve}\left [-\int _1^{\frac {y(x)}{x}}\frac {i \sqrt {3} \sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )-2 \operatorname {BesselJ}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) K[2]-2 \operatorname {BesselY}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 K[2]+i \sqrt {3} \operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}}{\left (\operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1\right ) K[2]^{3/2}}dK[2]-2 \log (x)+2 c_2=0,y(x)\right ]\] ✓ Maple : cpu = 0.245 (sec), leaf count = 166
dsolve(x^3*y(x)^2*diff(diff(y(x),x),x)+(y(x)+x)*(x*diff(y(x),x)-y(x))^3=0,y(x))
\[y \left (x \right ) = \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {i \sqrt {3}\, \operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}+i \sqrt {3}\, \operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}+\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}-2 c_{1} \operatorname {BesselY}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f} +\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}-2 \operatorname {BesselJ}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f}}{\textit {\_f}^{\frac {3}{2}} \left (\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1}+\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right )\right )}d \textit {\_f} \right )+2 c_{2}\right ) x\]