3.520   ODE No. 520

$$\boxed{{\displaystyle {\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^3+\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)-y\left(x\right)=0}}$$

1. Problem in Latex
2. Mathematica input
3. Maple input

Mathematica: cpu = 151.472735 (sec), leaf count = 3323 $$\{\{y(x)\to \text{InverseFunction}[\frac{(243{\mathrm{\#}\text{1}}^2-27\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-24\sqrt[3]{2}\sqrt[6]{3}{\tan }^{-1}(\frac{1}{\sqrt{3}}-{\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-24\sqrt[3]{2}\sqrt[6]{3}{\tan }^{-1}({\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+\frac{1}{\sqrt{3}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+8\sqrt[3]{2}{3}^{2/3}\log (6-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+8\sqrt[3]{2}{3}^{2/3}\log (2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-4\sqrt[3]{2}{3}^{2/3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-4\sqrt[3]{2}{3}^{2/3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+54){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{4/3}}{864\sqrt[3]{3}(-27{\mathrm{\#}\text{1}}^2+3\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-2)}+\frac{\sqrt{9{\mathrm{\#}\text{1}}^2+\frac{4}{3}}(22^{2/3}{3}^{5/6}{\tan }^{-1}(\frac{1}{\sqrt{3}}-{\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}})+22^{2/3}{3}^{5/6}{\tan }^{-1}({\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+\frac{1}{\sqrt{3}})+22^{2/3}\sqrt[3]{3}\log (6-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}})+22^{2/3}\sqrt[3]{3}\log (2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)-2^{2/3}\sqrt[3]{3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)-2^{2/3}\sqrt[3]{3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)+3{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3})(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}{36\sqrt[3]{2}(-27{\mathrm{\#}\text{1}}^2+3\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-4)}-\frac{(9\sqrt{3}\mathrm{\#}\text{1}+\sqrt{27{\mathrm{\#}\text{1}}^2+4})\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}}{323^{5/6}}-\frac{\log (\mathrm{\#}\text{1})}{36^{2/3}}\mathrm{\&}][c_1-\frac{x}{6^{2/3}}]\},\{y(x)\to \text{InverseFunction}[-\frac{i(-\frac{\sqrt[3]{3}(-i+\sqrt{3})(243{\mathrm{\#}\text{1}}^2-27\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-24\sqrt[3]{2}\sqrt[6]{3}{\tan }^{-1}(\frac{1}{\sqrt{3}}-{\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-24\sqrt[3]{2}\sqrt[6]{3}{\tan }^{-1}({\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+\frac{1}{\sqrt{3}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+8\sqrt[3]{2}{3}^{2/3}\log (6-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+8\sqrt[3]{2}{3}^{2/3}\log (2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-4\sqrt[3]{2}{3}^{2/3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-4\sqrt[3]{2}{3}^{2/3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+54){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{4/3}}{-27{\mathrm{\#}\text{1}}^2+3\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-2}+\frac{46^{2/3}(3+i\sqrt{3})\sqrt{27{\mathrm{\#}\text{1}}^2+4}(22^{2/3}{3}^{5/6}{\tan }^{-1}(\frac{1}{\sqrt{3}}-{\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}})+22^{2/3}{3}^{5/6}{\tan }^{-1}({\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+\frac{1}{\sqrt{3}})+22^{2/3}\sqrt[3]{3}\log (6-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}})+22^{2/3}\sqrt[3]{3}\log (2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)-2^{2/3}\sqrt[3]{3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)-2^{2/3}\sqrt[3]{3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)+3{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3})(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}{-27{\mathrm{\#}\text{1}}^2+3\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-4}+9\sqrt[3]{3}(3-i\sqrt{3})(9\sqrt{3}\mathrm{\#}\text{1}+\sqrt{27{\mathrm{\#}\text{1}}^2+4})\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+288i\sqrt[3]{2}\log (\mathrm{\#}\text{1}))}{34563^{5/6}}\mathrm{\&}][\frac{x}{22^{2/3}{3}^{5/6}}+c_1]\},\{y(x)\to \text{InverseFunction}\left[\frac{i(-\frac{\sqrt[3]{3}(i+\sqrt{3})(243{\mathrm{\#}\text{1}}^2-27\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-24\sqrt[3]{2}\sqrt[6]{3}{\tan }^{-1}(\frac{1}{\sqrt{3}}-{\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-24\sqrt[3]{2}\sqrt[6]{3}{\tan }^{-1}({\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+\frac{1}{\sqrt{3}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+8\sqrt[3]{2}{3}^{2/3}\log (6-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+8\sqrt[3]{2}{3}^{2/3}\log (2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-4\sqrt[3]{2}{3}^{2/3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-4\sqrt[3]{2}{3}^{2/3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+54){(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{4/3}}{-27{\mathrm{\#}\text{1}}^2+3\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-2}+\frac{46^{2/3}(3-i\sqrt{3})\sqrt{27{\mathrm{\#}\text{1}}^2+4}(22^{2/3}{3}^{5/6}{\tan }^{-1}(\frac{1}{\sqrt{3}}-{\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}})+22^{2/3}{3}^{5/6}{\tan }^{-1}({\left(\frac{2}{3}\right)}^{2/3}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+\frac{1}{\sqrt{3}})+22^{2/3}\sqrt[3]{3}\log (6-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}})+22^{2/3}\sqrt[3]{3}\log (2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)-2^{2/3}\sqrt[3]{3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}-2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)-2^{2/3}\sqrt[3]{3}\log (\sqrt[3]{2}{3}^{2/3}{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3}+2^{2/3}{3}^{5/6}\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}+6)+3{(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}^{2/3})(\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1})}{-27{\mathrm{\#}\text{1}}^2+3\sqrt{81{\mathrm{\#}\text{1}}^2+12}\mathrm{\#}\text{1}-4}+9\sqrt[3]{3}(3+i\sqrt{3})(9\sqrt{3}\mathrm{\#}\text{1}+\sqrt{27{\mathrm{\#}\text{1}}^2+4})\sqrt[3]{\sqrt{81{\mathrm{\#}\text{1}}^2+12}-9\mathrm{\#}\text{1}}-288i\sqrt[3]{2}\log (\mathrm{\#}\text{1}))}{34563^{5/6}}\mathrm{\&}\right][\frac{x}{22^{2/3}{3}^{5/6}}+c_1]\}\}$$

Maple: cpu = 0.062 (sec), leaf count = 245 $$\{x-{\int }^{y\left(x\right)}6\frac{\sqrt[3]{108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12}}}{{(108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12})}^{2/3}-12}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}12\frac{\sqrt[3]{108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12}}}{(i\sqrt{3}-1)(\sqrt[3]{108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12}}-\sqrt{3}-3i)(\sqrt[3]{108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12}}+\sqrt{3}+3i)}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}-12\frac{\sqrt[3]{108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12}}}{(i\sqrt{3}+1)(\sqrt[3]{108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12}}+\sqrt{3}-3i)(\sqrt[3]{108\mathit{\_}\mathit{a}+12\sqrt{81{\mathit{\_}\mathit{a}}^2+12}}-\sqrt{3}+3i)}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$