4.12.45 $\sqrt{x^2+1}(y(x)+1)y^{\prime }(x)=y(x{)}^3$

ODE
$$\sqrt{x^2+1}(y(x)+1)y^{\prime }(x)=y(x{)}^3$$ ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.0374284 (sec), leaf count = 63

$$\{\{y(x)\to -\frac{\sqrt{-2c_1-2{\sinh }^{-1}(x)+1}+1}{2(c_1+{\sinh }^{-1}(x))}\},\{y(x)\to \frac{\sqrt{-2c_1-2{\sinh }^{-1}(x)+1}-1}{2(c_1+{\sinh }^{-1}(x))}\}\}$$

Maple ✓
cpu = 0.013 (sec), leaf count = 17

$$\{\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(x\right)+{\left(y\left(x\right)\right)}^{-1}+\frac{1}{2{\left(y\left(x\right)\right)}^2}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[Sqrt[1 + x^2]*(1 + y[x])*y'[x] == y[x]^3,y[x],x]

Mathematica raw output

{{y[x] -> -(1 + Sqrt[1 - 2*ArcSinh[x] - 2*C[1]])/(2*(ArcSinh[x] + C[1]))}, {y[x]
 -> (-1 + Sqrt[1 - 2*ArcSinh[x] - 2*C[1]])/(2*(ArcSinh[x] + C[1]))}}

Maple raw input

dsolve((1+y(x))*diff(y(x),x)*(x^2+1)^(1/2) = y(x)^3, y(x),'implicit')

Maple raw output

arcsinh(x)+1/y(x)+1/2/y(x)^2+_C1 = 0