ODE
\[ y'(x)=x^{m-1} y(x)^{1-n} f\left (a x^m+b y(x)^n\right ) \] ODE Classification
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Book solution method
Change of Variable, new dependent variable
Mathematica ✓
cpu = 0.823114 (sec), leaf count = 170
\[\text {Solve}\left [c_1=\int _1^{y(x)}\left (-\frac {a m K[2]^{n-1}}{a m+b n f\left (a x^m+b K[2]^n\right )}-\int _1^x\frac {a^2 b m^2 n K[1]^{m-1} K[2]^{n-1} f'\left (a K[1]^m+b K[2]^n\right )}{\left (a m+b n f\left (a K[1]^m+b K[2]^n\right )\right )^2}dK[1]\right )dK[2]+\int _1^x\frac {a m f\left (a K[1]^m+b y(x)^n\right ) K[1]^{m-1}}{a m+b n f\left (a K[1]^m+b y(x)^n\right )}dK[1],y(x)\right ]\]
Maple ✓
cpu = 0.5 (sec), leaf count = 174
\[\left [y \left (x \right ) = \left (-\frac {-\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} b n \textit {\_a} -\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} a m n +a \,m^{2}}d \textit {\_a} \right ) b \,m^{2}+\textit {\_C1} m -x^{m}\right ) b +a \,x^{m}}{b}\right )^{\frac {1}{n}}\right ]\] Mathematica raw input
DSolve[y'[x] == x^(-1 + m)*f[a*x^m + b*y[x]^n]*y[x]^(1 - n),y[x],x]
Mathematica raw output
Solve[C[1] == Inactive[Integrate][(a*m*f[a*K[1]^m + b*y[x]^n]*K[1]^(-1 + m))/(a*m + b*n*f[a*K[1]^m + b*y[x]^n]), {K[1], 1, x}] + Inactive[Integrate][-((a*m*K[2]^(-1 + n))/(a*m + b*n*f[a*x^m + b*K[2]^n])) - Inactive[Integrate][(a^2*b*m^2*n*K[1]^(-1 + m)*K[2]^(-1 + n)*Derivative[1][f][a*K[1]^m + b*K[2]^n])/(a*m + b*n*f[a*K[1]^m + b*K[2]^n])^2, {K[1], 1, x}], {K[2], 1, y[x]}], y[x]]
Maple raw input
dsolve(diff(y(x),x) = x^(m-1)*y(x)^(1-n)*f(a*x^m+b*y(x)^n), y(x))
Maple raw output
[y(x) = (-1/b*(-RootOf(Intat(1/((m^(1/m))^m*f(a*(m^(1/m))^m+b*(((_a*b-a*m)/b)^(1/n))^n)*(((_a*b-a*m)/b)^(1/n))^(-n)*b*n*_a-(m^(1/m))^m*f(a*(m^(1/m))^m+b*(((_a*b-a*m)/b)^(1/n))^n)*(((_a*b-a*m)/b)^(1/n))^(-n)*a*m*n+a*m^2),_a = _Z)*b*m^2+_C1*m-x^m)*b+a*x^m))^(1/n)]