#### 2.538   ODE No. 538

$2 \left (x y'(x)+y(x)\right )^3-y(x) y'(x)=0$ Mathematica : cpu = 9.27258 (sec), leaf count = 110

$\left \{\left \{y(x)\to \frac {\int _1^x\frac {\text {InverseFunction}\left [-\frac {2 \sqrt {\text {\#1}^2-8 \text {\#1}^3} \tan ^{-1}\left (\sqrt {8 \text {\#1}-1}\right )}{\text {\#1} \sqrt {8 \text {\#1}-1}}+\log (8 \text {\#1}-1)+\log \left (1+\frac {1}{8 \text {\#1}-1}\right )+\frac {3 \sqrt {\text {\#1}^2-8 \text {\#1}^3}}{\text {\#1}}\& \right ][c_1+2 \log (\text {K\105714})]}{\text {K\105714}}d\text {K\105714}}{x}\right \}\right \}$ Maple : cpu = 1.213 (sec), leaf count = 1625

$\left \{ \int _{{\it \_b}}^{x}\!{ \left ( -{6}^{{\frac {2}{3}}} \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}+6\,{\it \_a}\,y \left ( x \right ) \left ( \sqrt [3]{6}\sqrt [3]{-9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2}}-1 \right ) \right ) \left ( 6\,{{\it \_a}}^{2}y \left ( x \right ) +{\it \_a}\,{6}^{{\frac {2}{3}}} \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!6\,{x\sqrt [3]{6}\sqrt [3]{-9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f}} \left ( {6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+6\,{\it \_f}\,x \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!324\,{{\it \_a}\,\sqrt [3]{6}{\it \_f} \left ( {{\it \_a}\,{\it \_f} \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}-2/3\,{\it \_a}\, \left ( -1/18\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{-2/3}}-1/6\, \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}+1/3\,{\it \_a}\, \left ( -2/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) {6}^{2/3} \right ) \left ( {6}^{2/3} \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+6\,{\it \_a}\,{\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{\frac {1}{{\it \_a}} \left ( \left ( -3\,i\sqrt [6]{3}{2}^{{\frac {2}{3}}}-{6}^{{\frac {2}{3}}} \right ) \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}+6\, \left ( i\sqrt {3}-2\,\sqrt [3]{6}\sqrt [3]{-9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2}}-1 \right ) y \left ( x \right ) {\it \_a} \right ) \left ( 3\,i\sqrt [6]{3}{2}^{{\frac {2}{3}}} \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}-6\,i\sqrt {3}{\it \_a}\,y \left ( x \right ) +{6}^{{\frac {2}{3}}} \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}+6\,{\it \_a}\,y \left ( x \right ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-12\,{x\sqrt [3]{6}\sqrt [3]{-9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f}} \left ( i{6}^{2/3}\sqrt {3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}-6\,i\sqrt {3}x{\it \_f}+{6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+6\,{\it \_f}\,x \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!18\,{{\it \_a}\,\sqrt [3]{6}{\it \_f} \left ( {{\it \_a}\, \left ( -1+i\sqrt {3} \right ) {\it \_f} \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}-2/3\,{\it \_a}\, \left ( -1/18\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{-2/3}}+1/2\, \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}+1/3\,{\it \_a}\, \left ( -2/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( i\sqrt [6]{3}{2}^{2/3}+1/3\,{6}^{2/3} \right ) \right ) \left ( \left ( -i/2{2}^{2/3}\sqrt [6]{3}-1/6\,{6}^{2/3} \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+{\it \_a}\, \left ( -1+i\sqrt {3} \right ) {\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{\frac {1}{{\it \_a}} \left ( \left ( -3\,i\sqrt [6]{3}{2}^{{\frac {2}{3}}}+{6}^{{\frac {2}{3}}} \right ) \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}+6\, \left ( i\sqrt {3}+2\,\sqrt [3]{6}\sqrt [3]{-9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2}}+1 \right ) y \left ( x \right ) {\it \_a} \right ) \left ( 3\,i\sqrt [6]{3}{2}^{{\frac {2}{3}}} \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}-6\,i\sqrt {3}{\it \_a}\,y \left ( x \right ) -{6}^{{\frac {2}{3}}} \left ( -9\, \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) y \left ( x \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}-6\,{\it \_a}\,y \left ( x \right ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!12\,{x\sqrt [3]{6}\sqrt [3]{-9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f}} \left ( i{6}^{2/3}\sqrt {3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}-6\,i\sqrt {3}x{\it \_f}-{6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}-6\,{\it \_f}\,x \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!-18\,{{\it \_a}\,\sqrt [3]{6}{\it \_f} \left ( {{\it \_a}\, \left ( 1+i\sqrt {3} \right ) {\it \_f} \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}-2/3\,{\it \_a}\, \left ( -1/18\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{-2/3}}+1/2\, \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}+1/3\,{\it \_a}\, \left ( -2/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( i\sqrt [6]{3}{2}^{2/3}-1/3\,{6}^{2/3} \right ) \right ) \left ( \left ( -i/2{2}^{2/3}\sqrt [6]{3}+1/6\,{6}^{2/3} \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+{\it \_a}\, \left ( 1+i\sqrt {3} \right ) {\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \}$