4.5.22 $ny(x)+x{y}^{\prime }(x)=f(x)g(x^{n}y(x))$

ODE
$$ny(x)+x{y}^{\prime }(x)=f(x)g(x^{n}y(x))$$ ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 4.29333 (sec), leaf count = 39

$$\text{Solve}[{\int }_1^{x}f(K[2])K[2{]}^{n-1}dK[2]+c_1={\int }_1^{x^{n}y(x)}\frac{1}{g(K[1])}dK[1],y(x)]$$

Maple ✓
cpu = 0.272 (sec), leaf count = 32

$$\{\int f\left(x\right)x^{n-1}\mathrm{d}x-{\int }^{x^{n}y\left(x\right)}{\left(g\left(\mathit{\_}\mathit{a}\right)\right)}^{-1}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[n*y[x] + x*y'[x] == f[x]*g[x^n*y[x]],y[x],x]

Mathematica raw output

Solve[C[1] + Integrate[f[K[2]]*K[2]^(-1 + n), {K[2], 1, x}] == Integrate[g[K[1]]
^(-1), {K[1], 1, x^n*y[x]}], y[x]]

Maple raw input

dsolve(x*diff(y(x),x)+n*y(x) = f(x)*g(x^n*y(x)), y(x),'implicit')

Maple raw output

Int(f(x)*x^(n-1),x)-Intat(1/g(_a),_a = x^n*y(x))-_C1 = 0