#### 2.322   ODE No. 322

$\left (10 x^2 y(x)^3-3 y(x)^2-2\right ) y'(x)+5 x y(x)^4+x=0$ Mathematica : cpu = 0.275024 (sec), leaf count = 2077

$\left \{\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}-\frac {1}{2} \sqrt {-\frac {\frac {32}{5 x^2}+\frac {8}{125 x^6}}{4 \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}}-\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}-\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {2}{25 x^4}}+\frac {1}{10 x^2}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}+\frac {1}{2} \sqrt {-\frac {\frac {32}{5 x^2}+\frac {8}{125 x^6}}{4 \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}}-\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}-\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {2}{25 x^4}}+\frac {1}{10 x^2}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}-\frac {1}{2} \sqrt {\frac {\frac {32}{5 x^2}+\frac {8}{125 x^6}}{4 \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}}-\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}-\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {2}{25 x^4}}+\frac {1}{10 x^2}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}+\frac {1}{2} \sqrt {\frac {\frac {32}{5 x^2}+\frac {8}{125 x^6}}{4 \sqrt {\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}+\frac {1}{25 x^4}}}-\frac {\sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}{15 \sqrt [3]{2} x^2}-\frac {4 \sqrt [3]{2} \left (5 x^4-10 c_1 x^2-2\right )}{5 x^2 \sqrt [3]{2268 x^2-216 c_1+\sqrt {\left (2160 x^2+108 \left (x^2-2 c_1\right )\right ){}^2-4 \left (60 x^4-120 c_1 x^2-24\right ){}^3}}}+\frac {2}{25 x^4}}+\frac {1}{10 x^2}\right \}\right \}$ Maple : cpu = 0.022 (sec), leaf count = 29

$\left \{ {\frac {5\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{4}}{2}}- \left ( y \left ( x \right ) \right ) ^{3}+{\frac {{x}^{2}}{2}}-2\,y \left ( x \right ) +{\it \_C1}=0 \right \}$