2.770   ODE No. 770

\[ y'(x)=\frac {2 y(x)^6}{32 x^2 y(x)^4+y(x)^3+16 x y(x)^2+2} \] Mathematica : cpu = 0.219852 (sec), leaf count = 705


\[\left \{\left \{y(x)\to \frac {\sqrt [3]{8192 x^3+18432 c_1{}^2 x^2+\sqrt {4 \left (-256 x^2+192 c_1{}^2 x-12 c_1\right ){}^3+\left (8192 x^3+18432 c_1{}^2 x^2-2880 c_1 x+108\right ){}^2}-2880 c_1 x+108}}{3 \sqrt [3]{2} (1-16 c_1 x)}-\frac {\sqrt [3]{2} \left (-256 x^2+192 c_1{}^2 x-12 c_1\right )}{3 (1-16 c_1 x) \sqrt [3]{8192 x^3+18432 c_1{}^2 x^2+\sqrt {4 \left (-256 x^2+192 c_1{}^2 x-12 c_1\right ){}^3+\left (8192 x^3+18432 c_1{}^2 x^2-2880 c_1 x+108\right ){}^2}-2880 c_1 x+108}}+\frac {16 x}{3 (1-16 c_1 x)}\right \},\left \{y(x)\to -\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{8192 x^3+18432 c_1{}^2 x^2+\sqrt {4 \left (-256 x^2+192 c_1{}^2 x-12 c_1\right ){}^3+\left (8192 x^3+18432 c_1{}^2 x^2-2880 c_1 x+108\right ){}^2}-2880 c_1 x+108}}{6 \sqrt [3]{2} (1-16 c_1 x)}+\frac {\left (1+i \sqrt {3}\right ) \left (-256 x^2+192 c_1{}^2 x-12 c_1\right )}{3\ 2^{2/3} (1-16 c_1 x) \sqrt [3]{8192 x^3+18432 c_1{}^2 x^2+\sqrt {4 \left (-256 x^2+192 c_1{}^2 x-12 c_1\right ){}^3+\left (8192 x^3+18432 c_1{}^2 x^2-2880 c_1 x+108\right ){}^2}-2880 c_1 x+108}}+\frac {16 x}{3 (1-16 c_1 x)}\right \},\left \{y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{8192 x^3+18432 c_1{}^2 x^2+\sqrt {4 \left (-256 x^2+192 c_1{}^2 x-12 c_1\right ){}^3+\left (8192 x^3+18432 c_1{}^2 x^2-2880 c_1 x+108\right ){}^2}-2880 c_1 x+108}}{6 \sqrt [3]{2} (1-16 c_1 x)}+\frac {\left (1-i \sqrt {3}\right ) \left (-256 x^2+192 c_1{}^2 x-12 c_1\right )}{3\ 2^{2/3} (1-16 c_1 x) \sqrt [3]{8192 x^3+18432 c_1{}^2 x^2+\sqrt {4 \left (-256 x^2+192 c_1{}^2 x-12 c_1\right ){}^3+\left (8192 x^3+18432 c_1{}^2 x^2-2880 c_1 x+108\right ){}^2}-2880 c_1 x+108}}+\frac {16 x}{3 (1-16 c_1 x)}\right \}\right \}\] Maple : cpu = 0.161 (sec), leaf count = 1105


\[y \relax (x ) = \frac {\left (4096 x^{3} c_{1}^{3}+6 \sqrt {3}\, \sqrt {4096 x^{3} c_{1}^{4}+2048 x^{2} c_{1}^{2}+576 x c_{1}^{3}+27 c_{1}^{4}+256 x +16 c_{1}}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 x^{3} c_{1}^{4}+2048 x^{2} c_{1}^{2}+576 x c_{1}^{3}+27 c_{1}^{4}+256 x +16 c_{1}}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 x^{2} c_{1}\right )^{\frac {1}{3}}}{3 c_{1}+48 x}+\frac {\frac {256}{3} x^{2} c_{1}^{2}-64 x -4 c_{1}}{\left (c_{1}+16 x \right ) \left (4096 x^{3} c_{1}^{3}+6 \sqrt {3}\, \sqrt {4096 x^{3} c_{1}^{4}+2048 x^{2} c_{1}^{2}+576 x c_{1}^{3}+27 c_{1}^{4}+256 x +16 c_{1}}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 x^{3} c_{1}^{4}+2048 x^{2} c_{1}^{2}+576 x c_{1}^{3}+27 c_{1}^{4}+256 x +16 c_{1}}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 x^{2} c_{1}\right )^{\frac {1}{3}}}+\frac {16 x c_{1}}{3 c_{1}+48 x}\]