##### 4.14.18 $$x \left (x y(x)^2+1\right ) y'(x)+y(x)=0$$

ODE
$x \left (x y(x)^2+1\right ) y'(x)+y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.330295 (sec), leaf count = 60

$\left \{\left \{y(x)\to \frac {1}{2} \left (c_1-\frac {\sqrt {4+c_1{}^2 x}}{\sqrt {x}}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\frac {\sqrt {4+c_1{}^2 x}}{\sqrt {x}}+c_1\right )\right \}\right \}$

Maple
cpu = 0.128 (sec), leaf count = 137

$\left [y \left (x \right ) = -\frac {\sqrt {-2 x \textit {\_C1} \left (-2 \textit {\_C1} -x +\sqrt {4 x \textit {\_C1} +x^{2}}\right )}}{2 x \textit {\_C1}}, y \left (x \right ) = \frac {\sqrt {-2 x \textit {\_C1} \left (-2 \textit {\_C1} -x +\sqrt {4 x \textit {\_C1} +x^{2}}\right )}}{2 x \textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {2}\, \sqrt {x \textit {\_C1} \left (2 \textit {\_C1} +x +\sqrt {4 x \textit {\_C1} +x^{2}}\right )}}{2 x \textit {\_C1}}, y \left (x \right ) = \frac {\sqrt {2}\, \sqrt {x \textit {\_C1} \left (2 \textit {\_C1} +x +\sqrt {4 x \textit {\_C1} +x^{2}}\right )}}{2 x \textit {\_C1}}\right ]$ Mathematica raw input

DSolve[y[x] + x*(1 + x*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (C[1] - Sqrt[4 + x*C[1]^2]/Sqrt[x])/2}, {y[x] -> (C[1] + Sqrt[4 + x*C[
1]^2]/Sqrt[x])/2}}

Maple raw input

dsolve(x*(1+x*y(x)^2)*diff(y(x),x)+y(x) = 0, y(x))

Maple raw output

[y(x) = -1/2/x/_C1*(-2*x*_C1*(-2*_C1-x+(4*_C1*x+x^2)^(1/2)))^(1/2), y(x) = 1/2/x
/_C1*(-2*x*_C1*(-2*_C1-x+(4*_C1*x+x^2)^(1/2)))^(1/2), y(x) = -1/2/x/_C1*2^(1/2)*
(x*_C1*(2*_C1+x+(4*_C1*x+x^2)^(1/2)))^(1/2), y(x) = 1/2/x/_C1*2^(1/2)*(x*_C1*(2*
_C1+x+(4*_C1*x+x^2)^(1/2)))^(1/2)]