#### 2.271   ODE No. 271

$\left (x^2+y(x)^2\right ) y'(x)+2 x (y(x)+2 x)=0$ Mathematica : cpu = 0.262278 (sec), leaf count = 370

$\left \{\left \{y(x)\to \frac {\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}\right \}\right \}$ Maple : cpu = 0.163 (sec), leaf count = 352

$\left \{ y \left ( x \right ) ={ \left ( {\frac {1}{2}\sqrt [3]{4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1}}}-2\,{\frac {{\it \_C1}\,{x}^{2}}{\sqrt [3]{4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1}}}} \right ) {\frac {1}{\sqrt {{\it \_C1}}}}},y \left ( x \right ) =-{\frac {1}{4} \left ( \left ( 4\,i{\it \_C1}\,{x}^{2}+i \left ( 4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}-4\,{\it \_C1}\,{x}^{2}+ \left ( 4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1}}}}{\frac {1}{\sqrt {{\it \_C1}}}}},y \left ( x \right ) ={\frac {1}{4} \left ( 4\,i\sqrt {3}{\it \_C1}\,{x}^{2}+i\sqrt {3} \left ( 4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1} \right ) ^{{\frac {2}{3}}}+4\,{\it \_C1}\,{x}^{2}- \left ( 4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{4-16\,{x}^{3}{{\it \_C1}}^{3/2}+4\,\sqrt {20\,{{\it \_C1}}^{3}{x}^{6}-8\,{x}^{3}{{\it \_C1}}^{3/2}+1}}}}{\frac {1}{\sqrt {{\it \_C1}}}}} \right \}$