ODE No. 538

$$2{(x{y}^{\prime }(x)+y(x))}^3-y(x)y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 7.85677 (sec), leaf count = 113

DSolve[-(y[x]*Derivative[1][y][x]) + 2*(y[x] + x*Derivative[1][y][x])^3 == 0,y[x],x]
 

$$\left\{\{y(x)\to \frac{{\int }_1^x\frac{\text{InverseFunction}[-\frac{2\sqrt{{\mathrm{\#}\text{1}}^2-8{\mathrm{\#}\text{1}}^3}{\tan }^{-1}\left(\sqrt{8\mathrm{\#}\text{1}-1}\right)}{\mathrm{\#}\text{1}\sqrt{8\mathrm{\#}\text{1}-1}}+\log (8\mathrm{\#}\text{1}-1)+\log (1+\frac{1}{8\mathrm{\#}\text{1}-1})+\frac{3\sqrt{{\mathrm{\#}\text{1}}^2-8{\mathrm{\#}\text{1}}^3}}{\mathrm{\#}\text{1}}\mathrm{\&}][c_1+2\log (K[1])]}{K[1]}dK[1]}{x}\}\right\}$$ ✓ Maple : cpu = 1.278 (sec), leaf count = 1625

dsolve(2*(x*diff(y(x),x)+y(x))^3-y(x)*diff(y(x),x)=0,y(x))
 

$${\int }_{\mathit{\text{\_b}}}^x\frac{-6^{\frac{2}{3}}{(-9y\left(x\right)(-\frac{\sqrt{3}\sqrt{\frac{27\mathit{\text{\_a}}y{\left(x\right)}^2-2y\left(x\right)}{\mathit{\text{\_a}}}}}{9}+y\left(x\right)){\mathit{\text{\_a}}}^2)}^{\frac{2}{3}}+6\mathit{\text{\_a}}y\left(x\right)(6^{\frac{1}{3}}{(-9y\left(x\right)(-\frac{\sqrt{3}\sqrt{\frac{27\mathit{\text{\_a}}y{\left(x\right)}^2-2y\left(x\right)}{\mathit{\text{\_a}}}}}{9}+y\left(x\right)){\mathit{\text{\_a}}}^2)}^{\frac{1}{3}}-1)}{6{\mathit{\text{\_a}}}^{2}y\left(x\right)+\mathit{\text{\_a}}{6}^{\frac{2}{3}}{(-9y\left(x\right)(-\frac{\sqrt{3}\sqrt{\frac{27\mathit{\text{\_a}}y{\left(x\right)}^2-2y\left(x\right)}{\mathit{\text{\_a}}}}}{9}+y\left(x\right)){\mathit{\text{\_a}}}^2)}^{\frac{2}{3}}}d\mathit{\text{\_a}}+{\int }_{}^{y\left(x\right)}(\frac{6x{6}^{\frac{1}{3}}{(-9x^2(-\frac{\sqrt{3}\sqrt{\frac{27{\mathit{\text{\_f}}}^{2}x-2\mathit{\text{\_f}}}{x}}}{9}+\mathit{\text{\_f}})\mathit{\text{\_f}})}^{\frac{1}{3}}}{6^{\frac{2}{3}}{(-9x^2(-\frac{\sqrt{3}\sqrt{\frac{27{\mathit{\text{\_f}}}^{2}x-2\mathit{\text{\_f}}}{x}}}{9}+\mathit{\text{\_f}})\mathit{\text{\_f}})}^{\frac{2}{3}}+6x\mathit{\text{\_f}}}-\left({\int }_{\mathit{\text{\_b}}}^x\frac{324\mathit{\text{\_a}}(\frac{(\frac{\mathit{\text{\_f}}\sqrt{3}(\mathit{\text{\_a}}\mathit{\text{\_f}}-\frac{1}{27})}{\sqrt{\frac{27\mathit{\text{\_a}}{\mathit{\text{\_f}}}^2-2\mathit{\text{\_f}}}{\mathit{\text{\_a}}}}}-\frac{2\mathit{\text{\_a}}(-\frac{\sqrt{3}\sqrt{\frac{27\mathit{\text{\_a}}{\mathit{\text{\_f}}}^2-2\mathit{\text{\_f}}}{\mathit{\text{\_a}}}}}{18}+\mathit{\text{\_f}})}{3})\mathit{\text{\_a}}\mathit{\text{\_f}}}{{(-9{\mathit{\text{\_a}}}^2(-\frac{\sqrt{3}\sqrt{\frac{27\mathit{\text{\_a}}{\mathit{\text{\_f}}}^2-2\mathit{\text{\_f}}}{\mathit{\text{\_a}}}}}{9}+\mathit{\text{\_f}})\mathit{\text{\_f}})}^{\frac{2}{3}}}-\frac{6^{\frac{2}{3}}(\frac{\mathit{\text{\_f}}\sqrt{3}(\mathit{\text{\_a}}\mathit{\text{\_f}}-\frac{1}{27})}{\sqrt{\frac{27\mathit{\text{\_a}}{\mathit{\text{\_f}}}^2-2\mathit{\text{\_f}}}{\mathit{\text{\_a}}}}}+\frac{\mathit{\text{\_a}}(-\frac{2\sqrt{3}\sqrt{\frac{27\mathit{\text{\_a}}{\mathit{\text{\_f}}}^2-2\mathit{\text{\_f}}}{\mathit{\text{\_a}}}}}{9}+\mathit{\text{\_f}})}{3})}{6})\mathit{\text{\_f}}{6}^{\frac{1}{3}}}{{(6^{\frac{2}{3}}{(-9{\mathit{\text{\_a}}}^2(-\frac{\sqrt{3}\sqrt{\frac{27\mathit{\text{\_a}}{\mathit{\text{\_f}}}^2-2\mathit{\text{\_f}}}{\mathit{\text{\_a}}}}}{9}+\mathit{\text{\_f}})\mathit{\text{\_f}})}^{\frac{2}{3}}+6\mathit{\text{\_a}}\mathit{\text{\_f}})}^2}d\mathit{\text{\_a}}\right))d\mathit{\text{\_f}}+c_1=0$$