#### 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 = 0.21323 (sec), leaf count = 705

$\left \{\left \{y(x)\to \frac {\sqrt [3]{18432 c_1{}^2 x^2+\sqrt {4 \left (192 c_1{}^2 x-12 c_1-256 x^2\right ){}^3+\left (18432 c_1{}^2 x^2-2880 c_1 x+8192 x^3+108\right ){}^2}-2880 c_1 x+8192 x^3+108}}{3 \sqrt [3]{2} (1-16 c_1 x)}-\frac {\sqrt [3]{2} \left (192 c_1{}^2 x-12 c_1-256 x^2\right )}{3 (1-16 c_1 x) \sqrt [3]{18432 c_1{}^2 x^2+\sqrt {4 \left (192 c_1{}^2 x-12 c_1-256 x^2\right ){}^3+\left (18432 c_1{}^2 x^2-2880 c_1 x+8192 x^3+108\right ){}^2}-2880 c_1 x+8192 x^3+108}}+\frac {16 x}{3 (1-16 c_1 x)}\right \},\left \{y(x)\to -\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{18432 c_1{}^2 x^2+\sqrt {4 \left (192 c_1{}^2 x-12 c_1-256 x^2\right ){}^3+\left (18432 c_1{}^2 x^2-2880 c_1 x+8192 x^3+108\right ){}^2}-2880 c_1 x+8192 x^3+108}}{6 \sqrt [3]{2} (1-16 c_1 x)}+\frac {\left (1+i \sqrt {3}\right ) \left (192 c_1{}^2 x-12 c_1-256 x^2\right )}{3\ 2^{2/3} (1-16 c_1 x) \sqrt [3]{18432 c_1{}^2 x^2+\sqrt {4 \left (192 c_1{}^2 x-12 c_1-256 x^2\right ){}^3+\left (18432 c_1{}^2 x^2-2880 c_1 x+8192 x^3+108\right ){}^2}-2880 c_1 x+8192 x^3+108}}+\frac {16 x}{3 (1-16 c_1 x)}\right \},\left \{y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{18432 c_1{}^2 x^2+\sqrt {4 \left (192 c_1{}^2 x-12 c_1-256 x^2\right ){}^3+\left (18432 c_1{}^2 x^2-2880 c_1 x+8192 x^3+108\right ){}^2}-2880 c_1 x+8192 x^3+108}}{6 \sqrt [3]{2} (1-16 c_1 x)}+\frac {\left (1-i \sqrt {3}\right ) \left (192 c_1{}^2 x-12 c_1-256 x^2\right )}{3\ 2^{2/3} (1-16 c_1 x) \sqrt [3]{18432 c_1{}^2 x^2+\sqrt {4 \left (192 c_1{}^2 x-12 c_1-256 x^2\right ){}^3+\left (18432 c_1{}^2 x^2-2880 c_1 x+8192 x^3+108\right ){}^2}-2880 c_1 x+8192 x^3+108}}+\frac {16 x}{3 (1-16 c_1 x)}\right \}\right \}$ Maple : cpu = 0.105 (sec), leaf count = 1105

$\left \{ y \left ( x \right ) ={\frac {1}{96\,x+6\,{\it \_C1}} \left ( 32\,{\it \_C1}\,x\sqrt [3]{96\, \left ( {\it \_C1}/16+x \right ) \sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}+ \left ( 4096\,{x}^{3}+54 \right ) {{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2}}+ \left ( -256\,i{x}^{2}{{\it \_C1}}^{2}+i \left ( 4096\,{{\it \_C1}}^{3}{x}^{3}+6\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}{\it \_C1}+96\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}x+54\,{{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2} \right ) ^{{\frac {2}{3}}}+192\,ix+12\,i{\it \_C1} \right ) \sqrt {3}-256\,{{\it \_C1}}^{2}{x}^{2}- \left ( 4096\,{{\it \_C1}}^{3}{x}^{3}+6\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}{\it \_C1}+96\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}x+54\,{{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2} \right ) ^{{\frac {2}{3}}}+192\,x+12\,{\it \_C1} \right ) {\frac {1}{\sqrt [3]{96\, \left ( {\it \_C1}/16+x \right ) \sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}+ \left ( 4096\,{x}^{3}+54 \right ) {{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2}}}}},y \left ( x \right ) =-{\frac {1}{96\,x+6\,{\it \_C1}} \left ( -32\,{\it \_C1}\,x\sqrt [3]{96\, \left ( {\it \_C1}/16+x \right ) \sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}+ \left ( 4096\,{x}^{3}+54 \right ) {{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2}}+ \left ( -256\,i{x}^{2}{{\it \_C1}}^{2}+i \left ( 4096\,{{\it \_C1}}^{3}{x}^{3}+6\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}{\it \_C1}+96\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}x+54\,{{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2} \right ) ^{{\frac {2}{3}}}+192\,ix+12\,i{\it \_C1} \right ) \sqrt {3}+256\,{{\it \_C1}}^{2}{x}^{2}+ \left ( 4096\,{{\it \_C1}}^{3}{x}^{3}+6\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}{\it \_C1}+96\,\sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}x+54\,{{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2} \right ) ^{{\frac {2}{3}}}-192\,x-12\,{\it \_C1} \right ) {\frac {1}{\sqrt [3]{96\, \left ( {\it \_C1}/16+x \right ) \sqrt {3}\sqrt { \left ( 4096\,{x}^{3}+27 \right ) {{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}+ \left ( 4096\,{x}^{3}+54 \right ) {{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2}}}}},y \left ( x \right ) ={\frac {1}{3\,{\it \_C1}+48\,x}\sqrt [3]{4096\,{{\it \_C1}}^{3}{x}^{3}+6\,\sqrt {3}\sqrt {4096\,{{\it \_C1}}^{4}{x}^{3}+27\,{{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}{\it \_C1}+96\,\sqrt {3}\sqrt {4096\,{{\it \_C1}}^{4}{x}^{3}+27\,{{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}x+54\,{{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2}}}+{\frac {256\,{{\it \_C1}}^{2}{x}^{2}-12\,{\it \_C1}-192\,x}{3\,{\it \_C1}+48\,x}{\frac {1}{\sqrt [3]{4096\,{{\it \_C1}}^{3}{x}^{3}+6\,\sqrt {3}\sqrt {4096\,{{\it \_C1}}^{4}{x}^{3}+27\,{{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}{\it \_C1}+96\,\sqrt {3}\sqrt {4096\,{{\it \_C1}}^{4}{x}^{3}+27\,{{\it \_C1}}^{4}+576\,x{{\it \_C1}}^{3}+2048\,{{\it \_C1}}^{2}{x}^{2}+16\,{\it \_C1}+256\,x}x+54\,{{\it \_C1}}^{3}+1440\,{{\it \_C1}}^{2}x+9216\,{\it \_C1}\,{x}^{2}}}}}+16\,{\frac {{\it \_C1}\,x}{3\,{\it \_C1}+48\,x}} \right \}$