4.14.44 $xy(x{)}^3{y}^{\prime }(x)=(1-x^2)(y(x{)}^2+1)$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.0237565 (sec), leaf count = 61

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

Maple ✓
cpu = 0.01 (sec), leaf count = 29

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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