ODE
\[ x y'(x) \left (a y(x)+x^n\right )+y(x)^2 (b+c y(x))=0 \] 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 = 2.87119 (sec), leaf count = 77
\[\text {Solve}\left [-\frac {x^{-n} y(x)^{-\frac {a n+b}{b}} (b+c y(x))^{\frac {a n}{b}} \left (y(x) \left (a^2 n+a b-c x^n\right )+a n x^n\right )}{a^2 n^2 (a n+b)}=c_1,y(x)\right ]\]
Maple ✓
cpu = 0.256 (sec), leaf count = 107
\[\left [y \left (x \right ) = \frac {b}{\RootOf \left (-x^{-n} \textit {\_Z}^{\frac {a n}{b}} a^{2} b n -x^{-n} \textit {\_Z}^{\frac {a n}{b}} a \,b^{2}+\textit {\_C1} \,a^{2} n^{2}+\textit {\_Z}^{\frac {a n}{b}} a c n -\textit {\_Z}^{\frac {a n +b}{b}} a n b +\textit {\_C1} a b n +\textit {\_Z}^{\frac {a n}{b}} b c \right ) b -c}\right ]\] 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))
Maple raw output
[y(x) = b/(RootOf(-x^(-n)*_Z^(a/b*n)*a^2*b*n-x^(-n)*_Z^(a/b*n)*a*b^2+_C1*a^2*n^2+_Z^(a/b*n)*a*c*n-_Z^((a*n+b)/b)*a*n*b+_C1*a*b*n+_Z^(a/b*n)*b*c)*b-c)]