#### 2.723   ODE No. 723

$y'(x)=\frac {2 a}{32 a^3 x^2-16 a^2 x y(x)^2+2 a y(x)^4+y(x)}$ Mathematica : cpu = 0.209268 (sec), leaf count = 663

$\left \{\left \{y(x)\to -\frac {\sqrt [3]{9216 a^5 c_1 x+\sqrt {4 \left (-192 a^3 x-64 a^4 c_1{}^2\right ){}^3+\left (9216 a^5 c_1 x-432 a^2-1024 a^6 c_1{}^3\right ){}^2}-432 a^2-1024 a^6 c_1{}^3}}{12 \sqrt [3]{2} a}+\frac {-192 a^3 x-64 a^4 c_1{}^2}{6\ 2^{2/3} a \sqrt [3]{9216 a^5 c_1 x+\sqrt {4 \left (-192 a^3 x-64 a^4 c_1{}^2\right ){}^3+\left (9216 a^5 c_1 x-432 a^2-1024 a^6 c_1{}^3\right ){}^2}-432 a^2-1024 a^6 c_1{}^3}}+\frac {2 a c_1}{3}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{9216 a^5 c_1 x+\sqrt {4 \left (-192 a^3 x-64 a^4 c_1{}^2\right ){}^3+\left (9216 a^5 c_1 x-432 a^2-1024 a^6 c_1{}^3\right ){}^2}-432 a^2-1024 a^6 c_1{}^3}}{24 \sqrt [3]{2} a}-\frac {\left (1+i \sqrt {3}\right ) \left (-192 a^3 x-64 a^4 c_1{}^2\right )}{12\ 2^{2/3} a \sqrt [3]{9216 a^5 c_1 x+\sqrt {4 \left (-192 a^3 x-64 a^4 c_1{}^2\right ){}^3+\left (9216 a^5 c_1 x-432 a^2-1024 a^6 c_1{}^3\right ){}^2}-432 a^2-1024 a^6 c_1{}^3}}+\frac {2 a c_1}{3}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{9216 a^5 c_1 x+\sqrt {4 \left (-192 a^3 x-64 a^4 c_1{}^2\right ){}^3+\left (9216 a^5 c_1 x-432 a^2-1024 a^6 c_1{}^3\right ){}^2}-432 a^2-1024 a^6 c_1{}^3}}{24 \sqrt [3]{2} a}-\frac {\left (1-i \sqrt {3}\right ) \left (-192 a^3 x-64 a^4 c_1{}^2\right )}{12\ 2^{2/3} a \sqrt [3]{9216 a^5 c_1 x+\sqrt {4 \left (-192 a^3 x-64 a^4 c_1{}^2\right ){}^3+\left (9216 a^5 c_1 x-432 a^2-1024 a^6 c_1{}^3\right ){}^2}-432 a^2-1024 a^6 c_1{}^3}}+\frac {2 a c_1}{3}\right \}\right \}$ Maple : cpu = 0.253 (sec), leaf count = 856

$\left \{ y \left ( x \right ) =-{\frac {1}{12\,a} \left ( -8\,{\it \_C1}\,{a}^{2}\sqrt [3]{ \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2}}+ \left ( -16\,i{{\it \_C1}}^{2}{a}^{4}-48\,i{a}^{3}x+i \left ( \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+16\,{{\it \_C1}}^{2}{a}^{4}+48\,{a}^{3}x+ \left ( \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{ \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2}}}}},y \left ( x \right ) ={\frac {1}{12\,a} \left ( 8\,{\it \_C1}\,{a}^{2}\sqrt [3]{ \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2}}+ \left ( -16\,i{{\it \_C1}}^{2}{a}^{4}-48\,i{a}^{3}x+i \left ( \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}-16\,{{\it \_C1}}^{2}{a}^{4}-48\,{a}^{3}x- \left ( \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{ \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2}}}}},y \left ( x \right ) ={\frac {1}{6\,a}\sqrt [3]{ \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2}}}+{\frac {8\,{a}^{2} \left ( a{{\it \_C1}}^{2}+3\,x \right ) }{3}{\frac {1}{\sqrt [3]{ \left ( 64\,{{\it \_C1}}^{3}{a}^{4}-576\,{\it \_C1}\,{a}^{3}x+3\,\sqrt {-12288\,{{\it \_C1}}^{4}{a}^{7}x+24576\,{{\it \_C1}}^{2}{a}^{6}{x}^{2}-12288\,{a}^{5}{x}^{3}+384\,{{\it \_C1}}^{3}{a}^{4}-3456\,{\it \_C1}\,{a}^{3}x+81}+27 \right ) {a}^{2}}}}}+{\frac {2\,{\it \_C1}\,a}{3}} \right \}$