#### 2.520   ODE No. 520

$y'(x)^3+y'(x)-y(x)=0$ Mathematica : cpu = 184.965 (sec), leaf count = 1590

$\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\left (27 \text {\#1}^2+4\right ) \left (27 \text {\#1}^2-3 \sqrt {81 \text {\#1}^2+12} \text {\#1}+4 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{12} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^2\right ) \left (-27 \text {\#1}^2+3 \sqrt {81 \text {\#1}^2+12} \text {\#1}-2\right )+6\right ) \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{4/3}}{48 \sqrt [3]{3} \left (27 \text {\#1}^2-3 \sqrt {81 \text {\#1}^2+12} \text {\#1}+4\right )^2}-\frac {\sqrt {9 \text {\#1}^2+\frac {4}{3}} \left (2\ 2^{2/3} 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{2} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}+\sqrt [3]{3}}{3^{5/6}}\right )-2\ 2^{2/3} \sqrt [3]{3} \log \left (2^{2/3} \sqrt [3]{3}-\left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}\right )+2^{2/3} \sqrt [3]{3} \log \left (\left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{4/3}+2^{2/3} \sqrt [3]{3} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}+2 \sqrt [3]{2} 3^{2/3}\right )-3 \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}\right ) \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )}{36 \sqrt [3]{2} \left (-27 \text {\#1}^2+3 \sqrt {81 \text {\#1}^2+12} \text {\#1}-4\right )}-\frac {\left (9 \sqrt {3} \text {\#1}+\sqrt {27 \text {\#1}^2+4}\right ) \sqrt [3]{\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}}}{32\ 3^{5/6}}-\frac {\log (\text {\#1})}{3\ 6^{2/3}}\& \right ]\left [c_1-\frac {x}{6^{2/3}}\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-i \left (\frac {\left (-i+\sqrt {3}\right ) \left (27 \text {\#1}^2+4\right ) \left (27 \text {\#1}^2-3 \sqrt {81 \text {\#1}^2+12} \text {\#1}+4 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{12} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^2\right ) \left (-27 \text {\#1}^2+3 \sqrt {81 \text {\#1}^2+12} \text {\#1}-2\right )+6\right ) \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{4/3}}{192 \sqrt {3} \left (27 \text {\#1}^2-3 \sqrt {81 \text {\#1}^2+12} \text {\#1}+4\right )^2}-\frac {i \left (-3 i+\sqrt {3}\right ) \sqrt {27 \text {\#1}^2+4} \left (2\ 2^{2/3} 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{2} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}+\sqrt [3]{3}}{3^{5/6}}\right )-2\ 2^{2/3} \sqrt [3]{3} \log \left (2^{2/3} \sqrt [3]{3}-\left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}\right )+2^{2/3} \sqrt [3]{3} \log \left (\left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{4/3}+2^{2/3} \sqrt [3]{3} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}+2 \sqrt [3]{2} 3^{2/3}\right )-3 \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}\right ) \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )}{432 \sqrt [3]{2} \sqrt [6]{3} \left (-27 \text {\#1}^2+3 \sqrt {81 \text {\#1}^2+12} \text {\#1}-4\right )}+\frac {\left (3-i \sqrt {3}\right ) \left (9 \sqrt {3} \text {\#1}+\sqrt {27 \text {\#1}^2+4}\right ) \sqrt [3]{\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}}}{384 \sqrt {3}}+\frac {i \log (\text {\#1})}{6\ 2^{2/3} 3^{5/6}}\right )\& \right ]\left [\frac {x}{2\ 2^{2/3} 3^{5/6}}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [i \left (\frac {\left (i+\sqrt {3}\right ) \left (27 \text {\#1}^2+4\right ) \left (27 \text {\#1}^2-3 \sqrt {81 \text {\#1}^2+12} \text {\#1}+4 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{12} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^2\right ) \left (-27 \text {\#1}^2+3 \sqrt {81 \text {\#1}^2+12} \text {\#1}-2\right )+6\right ) \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{4/3}}{192 \sqrt {3} \left (27 \text {\#1}^2-3 \sqrt {81 \text {\#1}^2+12} \text {\#1}+4\right )^2}+\frac {i \left (3 i+\sqrt {3}\right ) \sqrt {27 \text {\#1}^2+4} \left (2\ 2^{2/3} 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{2} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}+\sqrt [3]{3}}{3^{5/6}}\right )-2\ 2^{2/3} \sqrt [3]{3} \log \left (2^{2/3} \sqrt [3]{3}-\left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}\right )+2^{2/3} \sqrt [3]{3} \log \left (\left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{4/3}+2^{2/3} \sqrt [3]{3} \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}+2 \sqrt [3]{2} 3^{2/3}\right )-3 \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )^{2/3}\right ) \left (\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}\right )}{432 \sqrt [3]{2} \sqrt [6]{3} \left (-27 \text {\#1}^2+3 \sqrt {81 \text {\#1}^2+12} \text {\#1}-4\right )}+\frac {\left (3+i \sqrt {3}\right ) \left (9 \sqrt {3} \text {\#1}+\sqrt {27 \text {\#1}^2+4}\right ) \sqrt [3]{\sqrt {81 \text {\#1}^2+12}-9 \text {\#1}}}{384 \sqrt {3}}-\frac {i \log (\text {\#1})}{6\ 2^{2/3} 3^{5/6}}\right )\& \right ]\left [\frac {x}{2\ 2^{2/3} 3^{5/6}}+c_1\right ]\right \}\right \}$ Maple : cpu = 0.192 (sec), leaf count = 249

$\left \{ x-\int ^{y \left ( x \right ) }\!6\,{\frac {\sqrt [3]{108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12}}}{ \left ( 108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12} \right ) ^{2/3}-12}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!-12\,{\frac {\sqrt [3]{108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12}}}{ \left ( -1+i\sqrt {3} \right ) \left ( -\sqrt [3]{108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12}}+\sqrt {3}+3\,i \right ) \left ( \sqrt [3]{108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12}}+3\,i+\sqrt {3} \right ) }}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!12\,{\frac {\sqrt [3]{108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12}}}{ \left ( 1+i\sqrt {3} \right ) \left ( \sqrt [3]{108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12}}-\sqrt {3}+3\,i \right ) \left ( -\sqrt [3]{108\,{\it \_a}+12\,\sqrt {81\,{{\it \_a}}^{2}+12}}+3\,i-\sqrt {3} \right ) }}{d{\it \_a}}-{\it \_C1}=0 \right \}$