8.66   ODE No. 1656

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

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

Mathematica: cpu = 100.341242 (sec), leaf count = 33 $$\text{DSolve}[y^{″}(x)-a{(y^{\prime }(x{)}^2+1)}^{3/2}(bx+c+y(x))=0,y(x),x]$$

Maple: cpu = 0.499 (sec), leaf count = 771 $$\{y\left(x\right)=-bx+\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-x+{\int }^{\mathit{\_}\mathit{Z}}\frac{1}{({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)(b^2+1)b}(4a^2{b}^2{c}^2{\mathit{\_}\mathit{f}}^2+4a^2{b}^{2}c{\mathit{\_}\mathit{f}}^3+a^2{b}^2{\mathit{\_}\mathit{f}}^4-8\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{2}c\mathit{\_}\mathit{f}-4\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^2{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^2-2\sqrt{-b^2({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)}ac\mathit{\_}\mathit{f}-\sqrt{-b^2({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)}a{\mathit{\_}\mathit{f}}^2-4b^4+2\sqrt{-b^2({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)}\mathit{\_}\mathit{C}\mathit{1}a-4b^2)d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{2}),y\left(x\right)=-bx+\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-x+{\int }^{\mathit{\_}\mathit{Z}}-\frac{1}{({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)(b^2+1)b}(-a^2{b}^2{\mathit{\_}\mathit{f}}^4-4a^2{b}^{2}c{\mathit{\_}\mathit{f}}^3-4a^2{b}^2{c}^2{\mathit{\_}\mathit{f}}^2+4\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^2{\mathit{\_}\mathit{f}}^2+8\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{2}c\mathit{\_}\mathit{f}-4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^2-\sqrt{-b^2({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)}a{\mathit{\_}\mathit{f}}^2-2\sqrt{-b^2({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)}ac\mathit{\_}\mathit{f}+4b^4+2\sqrt{-b^2({\mathit{\_}\mathit{f}}^4{a}^2+4{\mathit{\_}\mathit{f}}^3{a}^{2}c+4{\mathit{\_}\mathit{f}}^2{a}^2{c}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{f}}^2{a}^2-8\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{f}{a}^{2}c+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-4b^2-4)}\mathit{\_}\mathit{C}\mathit{1}a+4b^2)d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{2})\}$$