ODE
\[ \sqrt {x^2+1} (y(x)+1) y'(x)=y(x)^3 \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.38074 (sec), leaf count = 63
\[\left \{\left \{y(x)\to -\frac {1+\sqrt {-2 \sinh ^{-1}(x)+1-2 c_1}}{2 \left (\sinh ^{-1}(x)+c_1\right )}\right \},\left \{y(x)\to \frac {-1+\sqrt {-2 \sinh ^{-1}(x)+1-2 c_1}}{2 \left (\sinh ^{-1}(x)+c_1\right )}\right \}\right \}\]
Maple ✓
cpu = 0.028 (sec), leaf count = 49
\[\left [y \left (x \right ) = \frac {-1+\sqrt {1-2 \textit {\_C1} -2 \arcsinh \left (x \right )}}{2 \arcsinh \left (x \right )+2 \textit {\_C1}}, y \left (x \right ) = -\frac {1+\sqrt {1-2 \textit {\_C1} -2 \arcsinh \left (x \right )}}{2 \left (\arcsinh \left (x \right )+\textit {\_C1} \right )}\right ]\] Mathematica raw input
DSolve[Sqrt[1 + x^2]*(1 + y[x])*y'[x] == y[x]^3,y[x],x]
Mathematica raw output
{{y[x] -> -1/2*(1 + Sqrt[1 - 2*ArcSinh[x] - 2*C[1]])/(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))
Maple raw output
[y(x) = 1/2*(-1+(1-2*_C1-2*arcsinh(x))^(1/2))/(arcsinh(x)+_C1), y(x) = -1/2*(1+(1-2*_C1-2*arcsinh(x))^(1/2))/(arcsinh(x)+_C1)]