#### 2.821   ODE No. 821

$y'(x)=\frac {y(x) (x y(x)+1)}{x \left (x^3 y(x)^4-x y(x)-1\right )}$ Mathematica : cpu = 0.234498 (sec), leaf count = 2093

$\left \{\left \{y(x)\to \frac {c_1}{4}-\frac {1}{2} \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}-\frac {1}{2} \sqrt {\frac {c_1{}^2}{2}-\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}-\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}-\frac {c_1{}^3-\frac {4}{x^2}}{4 \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}}}\right \},\left \{y(x)\to \frac {c_1}{4}-\frac {1}{2} \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}+\frac {1}{2} \sqrt {\frac {c_1{}^2}{2}-\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}-\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}-\frac {c_1{}^3-\frac {4}{x^2}}{4 \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}}}\right \},\left \{y(x)\to \frac {c_1}{4}+\frac {1}{2} \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}-\frac {1}{2} \sqrt {\frac {c_1{}^2}{2}-\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}-\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}+\frac {c_1{}^3-\frac {4}{x^2}}{4 \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}}}\right \},\left \{y(x)\to \frac {c_1}{4}+\frac {1}{2} \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}+\frac {1}{2} \sqrt {\frac {c_1{}^2}{2}-\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}-\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}+\frac {c_1{}^3-\frac {4}{x^2}}{4 \sqrt {\frac {c_1{}^2}{4}+\frac {\sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}{18 \sqrt [3]{2} x^3}+\frac {\sqrt [3]{2} \left (3 c_1 x^4+8 x^3\right )}{x^3 \sqrt [3]{1944 c_1{}^2 x^6+1458 x^5+\sqrt {\left (1944 c_1{}^2 x^6+1458 x^5\right ){}^2-4 \left (54 c_1 x^4+144 x^3\right ){}^3}}}}}}\right \}\right \}$ Maple : cpu = 0.122 (sec), leaf count = 27

$\left \{ -{\frac {1}{2\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{2}}}-{\frac {1}{3\,{x}^{3} \left ( y \left ( x \right ) \right ) ^{3}}}-y \left ( x \right ) +{\it \_C1}=0 \right \}$