ODE
\[ 2 x (x+1) y(x) y'(x)=y(x)^2+1 \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.403901 (sec), leaf count = 58
\[\left \{\left \{y(x)\to -\frac {\sqrt {-1+\left (-1+e^{2 c_1}\right ) x}}{\sqrt {x+1}}\right \},\left \{y(x)\to \frac {\sqrt {-1+\left (-1+e^{2 c_1}\right ) x}}{\sqrt {x+1}}\right \}\right \}\]
Maple ✓
cpu = 0.026 (sec), leaf count = 42
\[\left [y \left (x \right ) = \frac {\sqrt {\left (x +1\right ) \left (x \textit {\_C1} -1\right )}}{x +1}, y \left (x \right ) = -\frac {\sqrt {\left (x +1\right ) \left (x \textit {\_C1} -1\right )}}{x +1}\right ]\] Mathematica raw input
DSolve[2*x*(1 + x)*y[x]*y'[x] == 1 + y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[-1 + (-1 + E^(2*C[1]))*x]/Sqrt[1 + x])}, {y[x] -> Sqrt[-1 + (-1 + E^(2*C[1]))*x]/Sqrt[1 + x]}}
Maple raw input
dsolve(2*(x+1)*x*y(x)*diff(y(x),x) = 1+y(x)^2, y(x))
Maple raw output
[y(x) = 1/(x+1)*((x+1)*(_C1*x-1))^(1/2), y(x) = -1/(x+1)*((x+1)*(_C1*x-1))^(1/2)]