#### 2.327   ODE No. 327

$\left (2 x^2 y(x)^3+x y(x)^4+2 y(x)+x\right ) y'(x)+y(x)^5+y(x)=0$ Mathematica : cpu = 0.338353 (sec), leaf count = 669

$\left \{\left \{y(x)\to \frac {\sqrt [3]{27 x^2+9 c_1{}^2 x^2+3 \sqrt {3} \sqrt {27 x^4-4 c_1{}^3 x^6-c_1{}^4 x^4+18 c_1{}^2 x^4+4 c_1{}^3 x^2}+2 c_1{}^3}}{3 \sqrt [3]{2} x}-\frac {\sqrt [3]{2} \left (-3 c_1 x^2-c_1{}^2\right )}{3 x \sqrt [3]{27 x^2+9 c_1{}^2 x^2+3 \sqrt {3} \sqrt {27 x^4-4 c_1{}^3 x^6-c_1{}^4 x^4+18 c_1{}^2 x^4+4 c_1{}^3 x^2}+2 c_1{}^3}}+\frac {c_1}{3 x}\right \},\left \{y(x)\to -\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^2+9 c_1{}^2 x^2+3 \sqrt {3} \sqrt {27 x^4-4 c_1{}^3 x^6-c_1{}^4 x^4+18 c_1{}^2 x^4+4 c_1{}^3 x^2}+2 c_1{}^3}}{6 \sqrt [3]{2} x}+\frac {\left (1+i \sqrt {3}\right ) \left (-3 c_1 x^2-c_1{}^2\right )}{3\ 2^{2/3} x \sqrt [3]{27 x^2+9 c_1{}^2 x^2+3 \sqrt {3} \sqrt {27 x^4-4 c_1{}^3 x^6-c_1{}^4 x^4+18 c_1{}^2 x^4+4 c_1{}^3 x^2}+2 c_1{}^3}}+\frac {c_1}{3 x}\right \},\left \{y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^2+9 c_1{}^2 x^2+3 \sqrt {3} \sqrt {27 x^4-4 c_1{}^3 x^6-c_1{}^4 x^4+18 c_1{}^2 x^4+4 c_1{}^3 x^2}+2 c_1{}^3}}{6 \sqrt [3]{2} x}+\frac {\left (1-i \sqrt {3}\right ) \left (-3 c_1 x^2-c_1{}^2\right )}{3\ 2^{2/3} x \sqrt [3]{27 x^2+9 c_1{}^2 x^2+3 \sqrt {3} \sqrt {27 x^4-4 c_1{}^3 x^6-c_1{}^4 x^4+18 c_1{}^2 x^4+4 c_1{}^3 x^2}+2 c_1{}^3}}+\frac {c_1}{3 x}\right \}\right \}$ Maple : cpu = 0.128 (sec), leaf count = 583

$\left \{ y \left ( x \right ) ={\frac {1}{12\,{\it \_C1}\,x} \left ( \left ( -12\,i{x}^{2}{\it \_C1}-i \left ( 108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+18\,{{\it \_C1}}^{2}{x}^{2}+ \left ( 4\,{x}^{4}-4 \right ) {\it \_C1}-{x}^{2}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8 \right ) ^{{\frac {2}{3}}}+4\,i \right ) \sqrt {3}+12\,{\it \_C1}\,{x}^{2}- \left ( \sqrt [3]{108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+18\,{{\it \_C1}}^{2}{x}^{2}+ \left ( 4\,{x}^{4}-4 \right ) {\it \_C1}-{x}^{2}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8}+2 \right ) ^{2} \right ) {\frac {1}{\sqrt [3]{108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+18\,{{\it \_C1}}^{2}{x}^{2}+ \left ( 4\,{x}^{4}-4 \right ) {\it \_C1}-{x}^{2}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8}}}},y \left ( x \right ) ={\frac {1}{12\,{\it \_C1}\,x} \left ( \left ( 12\,i{x}^{2}{\it \_C1}+i \left ( 108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+18\,{{\it \_C1}}^{2}{x}^{2}+ \left ( 4\,{x}^{4}-4 \right ) {\it \_C1}-{x}^{2}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8 \right ) ^{{\frac {2}{3}}}-4\,i \right ) \sqrt {3}+12\,{\it \_C1}\,{x}^{2}- \left ( \sqrt [3]{108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+18\,{{\it \_C1}}^{2}{x}^{2}+ \left ( 4\,{x}^{4}-4 \right ) {\it \_C1}-{x}^{2}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8}+2 \right ) ^{2} \right ) {\frac {1}{\sqrt [3]{108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+18\,{{\it \_C1}}^{2}{x}^{2}+ \left ( 4\,{x}^{4}-4 \right ) {\it \_C1}-{x}^{2}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8}}}},y \left ( x \right ) ={\frac {1}{6\,{\it \_C1}\,x}\sqrt [3]{108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+4\,{\it \_C1}\,{x}^{4}+18\,{{\it \_C1}}^{2}{x}^{2}-{x}^{2}-4\,{\it \_C1}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8}}-{\frac {6\,{\it \_C1}\,{x}^{2}-2}{3\,{\it \_C1}\,x}{\frac {1}{\sqrt [3]{108\,{{\it \_C1}}^{3}{x}^{2}+12\,\sqrt {3}\sqrt {27\,{{\it \_C1}}^{4}{x}^{2}+4\,{\it \_C1}\,{x}^{4}+18\,{{\it \_C1}}^{2}{x}^{2}-{x}^{2}-4\,{\it \_C1}}x{\it \_C1}+36\,{\it \_C1}\,{x}^{2}-8}}}}-{\frac {1}{3\,{\it \_C1}\,x}} \right \}$