8.132   ODE No. 1722

$$\boxed{{\displaystyle \left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)\right)y\left(x\right)-{\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2-1-2ay\left(x\right){({\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2+1)}^{3/2}=0}}$$

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

Mathematica: cpu = 1.926745 (sec), leaf count = 797 $$\{\{y(x)\to \text{InverseFunction}[-\frac{((4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1)E(i{\sinh }^{-1}\left(\sqrt{2}\sqrt{\frac{a^2}{-4c_1{a}^2+\sqrt{8c_1{a}^2+1}-1}}\mathrm{\#}\text{1}\right)|\frac{4c_1{a}^2-\sqrt{8c_1{a}^2+1}+1}{4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1})-(\sqrt{8c_1{a}^2+1}+1)F(i{\sinh }^{-1}\left(\sqrt{2}\sqrt{\frac{a^2}{-4c_1{a}^2+\sqrt{8c_1{a}^2+1}-1}}\mathrm{\#}\text{1}\right)|\frac{4c_1{a}^2-\sqrt{8c_1{a}^2+1}+1}{4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1}))\sqrt{1-\frac{2a^2{\mathrm{\#}\text{1}}^2}{4c_1{a}^2-\sqrt{8c_1{a}^2+1}+1}}\sqrt{1-\frac{2a^2{\mathrm{\#}\text{1}}^2}{4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1}}}{2a\sqrt{\frac{a^2}{-4c_1{a}^2+\sqrt{8c_1{a}^2+1}-1}}\sqrt{2a^2{\mathrm{\#}\text{1}}^4-2(4c_1{a}^2+1){\mathrm{\#}\text{1}}^2+8a^2{c}_1^2}}\mathrm{\&}][x+c_2]\},\{y(x)\to \text{InverseFunction}\left[\frac{((4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1)E(i{\sinh }^{-1}\left(\sqrt{2}\sqrt{\frac{a^2}{-4c_1{a}^2+\sqrt{8c_1{a}^2+1}-1}}\mathrm{\#}\text{1}\right)|\frac{4c_1{a}^2-\sqrt{8c_1{a}^2+1}+1}{4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1})-(\sqrt{8c_1{a}^2+1}+1)F(i{\sinh }^{-1}\left(\sqrt{2}\sqrt{\frac{a^2}{-4c_1{a}^2+\sqrt{8c_1{a}^2+1}-1}}\mathrm{\#}\text{1}\right)|\frac{4c_1{a}^2-\sqrt{8c_1{a}^2+1}+1}{4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1}))\sqrt{1-\frac{2a^2{\mathrm{\#}\text{1}}^2}{4c_1{a}^2-\sqrt{8c_1{a}^2+1}+1}}\sqrt{1-\frac{2a^2{\mathrm{\#}\text{1}}^2}{4c_1{a}^2+\sqrt{8c_1{a}^2+1}+1}}}{2a\sqrt{\frac{a^2}{-4c_1{a}^2+\sqrt{8c_1{a}^2+1}-1}}\sqrt{2a^2{\mathrm{\#}\text{1}}^4-2(4c_1{a}^2+1){\mathrm{\#}\text{1}}^2+8a^2{c}_1^2}}\mathrm{\&}\right][x+c_2]\}\}$$

Maple: cpu = 3.339 (sec), leaf count = 98 $$\{{\int }^{y\left(x\right)}({\mathit{\_}\mathit{a}}^{2}a+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{-{\mathit{\_}\mathit{a}}^4{a}^2-2\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{a}}^{2}a-{\mathit{\_}\mathit{C}\mathit{1}}^2+{\mathit{\_}\mathit{a}}^2}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-({\mathit{\_}\mathit{a}}^{2}a+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{-{\mathit{\_}\mathit{a}}^4{a}^2-2\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{a}}^{2}a-{\mathit{\_}\mathit{C}\mathit{1}}^2+{\mathit{\_}\mathit{a}}^2}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$