##### 4.14.37 $$y(x) \left (2 y(x)^2+1\right ) y'(x)=x \left (2 x^2+1\right )$$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.340394 (sec), leaf count = 151

$\left \{\left \{y(x)\to -\frac {\sqrt {-1-\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-1-\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {-1+\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-1+\sqrt {4 x^4+4 x^2+1+8 c_1}}}{\sqrt {2}}\right \}\right \}$

Maple
cpu = 0.047 (sec), leaf count = 113

$\left [y \left (x \right ) = -\frac {\sqrt {-2-2 \sqrt {4 x^{4}+4 x^{2}+8 \textit {\_C1} +1}}}{2}, y \left (x \right ) = \frac {\sqrt {-2-2 \sqrt {4 x^{4}+4 x^{2}+8 \textit {\_C1} +1}}}{2}, y \left (x \right ) = -\frac {\sqrt {-2+2 \sqrt {4 x^{4}+4 x^{2}+8 \textit {\_C1} +1}}}{2}, y \left (x \right ) = \frac {\sqrt {-2+2 \sqrt {4 x^{4}+4 x^{2}+8 \textit {\_C1} +1}}}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(Sqrt[-1 - Sqrt[1 + 4*x^2 + 4*x^4 + 8*C[1]]]/Sqrt[2])}, {y[x] -> Sqrt
[-1 - Sqrt[1 + 4*x^2 + 4*x^4 + 8*C[1]]]/Sqrt[2]}, {y[x] -> -(Sqrt[-1 + Sqrt[1 +
4*x^2 + 4*x^4 + 8*C[1]]]/Sqrt[2])}, {y[x] -> Sqrt[-1 + Sqrt[1 + 4*x^2 + 4*x^4 +
8*C[1]]]/Sqrt[2]}}

Maple raw input

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

Maple raw output

[y(x) = -1/2*(-2-2*(4*x^4+4*x^2+8*_C1+1)^(1/2))^(1/2), y(x) = 1/2*(-2-2*(4*x^4+4
*x^2+8*_C1+1)^(1/2))^(1/2), y(x) = -1/2*(-2+2*(4*x^4+4*x^2+8*_C1+1)^(1/2))^(1/2)
, y(x) = 1/2*(-2+2*(4*x^4+4*x^2+8*_C1+1)^(1/2))^(1/2)]