#### 2.243   ODE No. 243

$x (2 y(x)+x-1) y'(x)-y(x) (y(x)+2 x+1)=0$ Mathematica : cpu = 11.1893 (sec), leaf count = 487

$\left \{\left \{y(x)\to -\frac {\sqrt [3]{2} x}{\sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}+\frac {\sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{3 \sqrt [3]{2} c_1}+\frac {c_1 x-c_1}{c_1}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}+\frac {c_1 x-c_1}{c_1}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}+\frac {c_1 x-c_1}{c_1}\right \}\right \}$ Maple : cpu = 0.109 (sec), leaf count = 391

$\left \{ y \left ( x \right ) ={\frac {1}{80\,{\it \_C1}} \left ( -3\, \left ( x \left ( \sqrt {5}\sqrt {{\frac {80\, \left ( x-1 \right ) ^{2}{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2} \right ) ^{2/3} \left ( 1+i\sqrt {3} \right ) \sqrt [3]{5}+3\, \left ( x \left ( i\sqrt {3}-1 \right ) {5}^{2/3}+{\frac {80\,x-80}{3}\sqrt [3]{x \left ( \sqrt {5}\sqrt {{\frac {80\, \left ( x-1 \right ) ^{2}{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2}}} \right ) {\it \_C1} \right ) {\frac {1}{\sqrt [3]{x \left ( \sqrt {5}\sqrt {{\frac {80\, \left ( x-1 \right ) ^{2}{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2}}}}},y \left ( x \right ) ={\frac {1}{80\,{\it \_C1}} \left ( 3\, \left ( x \left ( \sqrt {5}\sqrt {{\frac {80\, \left ( x-1 \right ) ^{2}{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2} \right ) ^{2/3} \left ( i\sqrt {3}-1 \right ) \sqrt [3]{5}-3\, \left ( x \left ( 1+i\sqrt {3} \right ) {5}^{2/3}-{\frac {80\,x-80}{3}\sqrt [3]{x \left ( \sqrt {5}\sqrt {{\frac {80\, \left ( x-1 \right ) ^{2}{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2}}} \right ) {\it \_C1} \right ) {\frac {1}{\sqrt [3]{x \left ( \sqrt {5}\sqrt {{\frac {80\, \left ( x-1 \right ) ^{2}{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2}}}}},y \left ( x \right ) ={\frac {3\,\sqrt [3]{5}}{40\,{\it \_C1}}\sqrt [3]{x \left ( \sqrt {5}\sqrt {{\frac {80\,{\it \_C1}\,{x}^{2}-160\,{\it \_C1}\,x+80\,{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2}}}+{\frac {3\,x{5}^{2/3}}{40}{\frac {1}{\sqrt [3]{x \left ( \sqrt {5}\sqrt {{\frac {80\,{\it \_C1}\,{x}^{2}-160\,{\it \_C1}\,x+80\,{\it \_C1}-x}{{\it \_C1}}}}+20\,x-20 \right ) {{\it \_C1}}^{2}}}}}+x-1 \right \}$