4.15.38 $(x\sqrt{x^2+y(x{)}^2+1}-y(x)(x^2+y(x{)}^2))y^{\prime }(x)=\sqrt{x^2+y(x{)}^2+1}y(x)+x(x^2+y(x{)}^2)$

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

[[_1st_order, _with_linear_symmetries]]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.126706 (sec), leaf count = 27

$$\text{Solve}[\sqrt{x^2+y(x{)}^2+1}+{\tan }^{-1}\left(\frac{x}{y(x)}\right)=c_1,y(x)]$$

Maple ✓
cpu = 0.167 (sec), leaf count = 27

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

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

Mathematica raw output

Solve[ArcTan[x/y[x]] + Sqrt[1 + x^2 + y[x]^2] == C[1], y[x]]

Maple raw input

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

Maple raw output

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