4.41.31 $y(x)(y(x{)}^2+1)y^{″}(x)+(1-3y(x{)}^2)y^{\prime }(x{)}^2=0$

ODE
$$y(x)(y(x{)}^2+1)y^{″}(x)+(1-3y(x{)}^2)y^{\prime }(x{)}^2=0$$ ODE Classification

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.618433 (sec), leaf count = 78

$$\{\{y(x)\to -\frac{\sqrt{-2c_{1}x-2c_2{c}_1-1}}{\sqrt{2}\sqrt{c_1(c_2+x)}}\},\{y(x)\to \frac{\sqrt{-2c_{1}x-2c_2{c}_1-1}}{\sqrt{2}\sqrt{c_1(c_2+x)}}\}\}$$

Maple ✓
cpu = 0.13 (sec), leaf count = 23

$$\{-{(2{\left(y\left(x\right)\right)}^2+2)}^{-1}-\mathit{\_}\mathit{C}\mathit{1}x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[(1 - 3*y[x]^2)*y'[x]^2 + y[x]*(1 + y[x]^2)*y''[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(y(x)*(1+y(x)^2)*diff(diff(y(x),x),x)+(1-3*y(x)^2)*diff(y(x),x)^2 = 0, y(x),'implicit')

Maple raw output

-1/(2*y(x)^2+2)-_C1*x-_C2 = 0