\[ \boxed { y \left ( x \right ) \left ( \left ( y \left ( x \right ) \right ) ^{3}-2\,{x}^{3} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( 2\, \left ( y \left ( x \right ) \right ) ^{3}-{x}^{3} \right ) x=0} \]
Mathematica: cpu = 0.049006 (sec), leaf count = 139 \[ \text {Solve}\left [\frac {1}{7} \text {RootSum}\left [\text {$\#$1}^4+\text {$\#$1}^3+3 \text {$\#$1}^2+\text {$\#$1}+1\& ,\frac {8 \text {$\#$1}^3 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+9 \text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+12 \text {$\#$1} \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{4 \text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+1}\& \right ]-\frac {1}{7} \log \left (1-\frac {y(x)}{x}\right )=c_1-\log (x),y(x)\right ] \]
Maple: cpu = 0.296 (sec), leaf count = 1192 \[ \left \{ y \left ( x \right ) =-{\frac {\sqrt {3}x}{6} \left ( \sqrt {3}- 3\,{\it RootOf} \left ( 3\,{\it \_Z}-\sqrt {3}-\sqrt {3}{{\it \_Z}}^{2} +4\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) +3\,{{ \it \_Z}}^{3}+3\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {{\it \_Z}}^{4}+7\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {{\it \_Z}}^{2}-4\,{\it \_C1}+4\,\tan \left ( 1/6\, \sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\, \sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36 \,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) +\tan \left ( 7/6 \,\sqrt {3}\ln \left ( 2 \right ) \right ) \tan \left ( 1/6\,\sqrt {3} \ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{ {\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z }}^{2}-19 \right ) ^{2}}} \right ) \right ) \sqrt {3}{{\it \_Z}}^{2}- \tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {\it \_C1}\, \sqrt {3}{{\it \_Z}}^{2}+3\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {\it \_C1}\,\tan \left ( 1/6\,\sqrt {3}\ln \left ( { \frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3 }-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {{\it \_Z}}^{4}-\tan \left ( 1/6\, \sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\, \sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36 \,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {\it \_C1}\, \sqrt {3}{{\it \_Z}}^{2}+7\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {\it \_C1}\,\tan \left ( 1/6\,\sqrt {3}\ln \left ( { \frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3 }-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {{\it \_Z}}^{2}-7\,{{\it \_Z}}^{2}{ \it \_C1}-3\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) \tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x }^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {{ \it \_Z}}^{3}+\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) \tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+ 14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) \sqrt {3}-\sqrt {3}\tan \left ( 1/6\,\sqrt {3}\ln \left ( { \frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3 }-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {\it \_Z}-3\,\tan \left ( 7/6\,\sqrt { 3}\ln \left ( 2 \right ) \right ) \tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{ \it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z} }^{2}-19 \right ) ^{2}}} \right ) \right ) {\it \_Z}-\sqrt {3}\tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7 } \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{ \it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {{\it \_Z}}^{3}-\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) { \it \_C1}\,\sqrt {3}+4\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) \tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {\it \_C1}+3\,\tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{ 3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {\it \_C1}\,{{\it \_Z}}^{3}-\tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7 } \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{ \it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {\it \_C1}\,\sqrt {3}+3\,\tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac { \sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{ \it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{ 2}}} \right ) \right ) {\it \_C1}\,{\it \_Z}+3\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {\it \_C1}\,{{\it \_Z}}^{3}+\sqrt {3 }{\it \_C1}\,{\it \_Z}+3\,\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {\it \_C1}\,{\it \_Z}+\sqrt {3}{\it \_C1}\,{{\it \_Z }}^{3}-\sqrt {3}\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {\it \_Z}-\sqrt {3}\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) {{\it \_Z}}^{3}-\sqrt {3}\tan \left ( 7/6\,\sqrt {3} \ln \left ( 2 \right ) \right ) {\it \_C1}\,\tan \left ( 1/6\,\sqrt {3} \ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{ {\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z }}^{2}-19 \right ) ^{2}}} \right ) \right ) {{\it \_Z}}^{3}+3\,\tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{\it \_Z}-3}{{x}^{7 } \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^{4}+14\,\sqrt {3}{ \it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {{\it \_Z}}^{4}+7\,\tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{ \it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^ {4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {{\it \_Z}}^{2}-3\,{\it \_C1}\,{{\it \_Z}}^{4}- \sqrt {3}\tan \left ( 7/6\,\sqrt {3}\ln \left ( 2 \right ) \right ) { \it \_C1}\,\tan \left ( 1/6\,\sqrt {3}\ln \left ( {\frac {\sqrt {3}{ \it \_Z}-3}{{x}^{7} \left ( 6\,\sqrt {3}{{\it \_Z}}^{3}-9\,{{\it \_Z}}^ {4}+14\,\sqrt {3}{\it \_Z}-36\,{{\it \_Z}}^{2}-19 \right ) ^{2}}} \right ) \right ) {\it \_Z} \right ) \right ) } \right \} \]