##### 4.15.32 $$\left (x^2+1\right )^{3/2} \left (y(x)+\sqrt {y(x)^2+1}\right ) y'(x)=y(x)^2+1$$

ODE
$\left (x^2+1\right )^{3/2} \left (y(x)+\sqrt {y(x)^2+1}\right ) y'(x)=y(x)^2+1$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.39607 (sec), leaf count = 96

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

Maple
cpu = 0.11 (sec), leaf count = 28

$\left [\frac {x}{\sqrt {x^{2}+1}}-\arcsinh \left (y \left (x \right )\right )-\frac {\ln \left (1+y \left (x \right )^{2}\right )}{2}+\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[x/(x^2+1)^(1/2)-arcsinh(y(x))-1/2*ln(1+y(x)^2)+_C1 = 0]