4.12.17 $x{y}^{\prime }(x)(ay(x)+x^n)+y(x{)}^2(b+cy(x))=0$

ODE
$$x{y}^{\prime }(x)(ay(x)+x^n)+y(x{)}^2(b+cy(x))=0$$ ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 2.35039 (sec), leaf count = 77

$$\text{Solve}[-\frac{x^{-n}y(x{)}^{-\frac{an+b}{b}}(b+cy(x){)}^{\frac{an}{b}}(y(x)(a^{2}n+ab-c{x}^n)+an{x}^n)}{a^2{n}^2(an+b)}=c_1,y(x)]$$

Maple ✓
cpu = 0.055 (sec), leaf count = 77

$$\{\mathit{\_}\mathit{C}\mathit{1}+\frac{c-x^{-n}ab}{an}{\left(\frac{b+cy\left(x\right)}{by\left(x\right)}\right)}^{\frac{an}{b}}-\frac{b}{an+b}{\left(\frac{b+cy\left(x\right)}{by\left(x\right)}\right)}^{\frac{an}{b}+1}=0\}$$ Mathematica raw input

DSolve[y[x]^2*(b + c*y[x]) + x*(x^n + a*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[-(((b + c*y[x])^((a*n)/b)*(a*n*x^n + (a*b + a^2*n - c*x^n)*y[x]))/(a^2*n^2
*(b + a*n)*x^n*y[x]^((b + a*n)/b))) == C[1], y[x]]

Maple raw input

dsolve(x*(x^n+a*y(x))*diff(y(x),x)+(b+c*y(x))*y(x)^2 = 0, y(x),'implicit')

Maple raw output

_C1+((b+c*y(x))/y(x)/b)^(a/b*n)/a/n*(c-x^(-n)*a*b)-((b+c*y(x))/y(x)/b)^(a/b*n+1)
*b/(a*n+b) = 0