\[ f(x)^{1-n} g'(x) y(x)^n \left (-(a g(x)+b)^{-n}\right )-\frac {y(x) f'(x)}{f(x)}-f(x) g'(x)+y'(x)=0 \] ✓ Mathematica : cpu = 0.679161 (sec), leaf count = 96
\[\text {Solve}\left [\int _1^{\left (f(x)^{-n} (b+a g(x))^{-n}\right )^{\frac {1}{n}} y(x)}\frac {1}{K[1]^n-\left (a^n\right )^{\frac {1}{n}} K[1]+1}dK[1]=\frac {f(x) (a g(x)+b) \log (a g(x)+b) \left (f(x)^{-n} (a g(x)+b)^{-n}\right )^{\frac {1}{n}}}{a}+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.181 (sec), leaf count = 281
\[\left \{y \left (x \right ) = \frac {\left (a g \left (x \right )+b \right ) \RootOf \left (c_{1}-\left (\int _{}^{\textit {\_Z}}\frac {n^{-n} \left (\left (\frac {d}{d x}g \left (x \right )\right ) f \left (x \right )\right )^{-2 n +1} \left (\left (a g \left (x \right )+b \right )^{-n} f \left (x \right )^{-n +1} \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-n -1} \left (a n \left (a g \left (x \right )+b \right )^{-n -1} f \left (x \right )^{-n +2} \left (\frac {d}{d x}g \left (x \right )\right )^{3}\right )^{n}}{\textit {\_a} \,n^{-n} \left (\left (\frac {d}{d x}g \left (x \right )\right ) f \left (x \right )\right )^{-2 n +1} \left (\left (a g \left (x \right )+b \right )^{-n} f \left (x \right )^{-n +1} \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-n -1} \left (a n \left (a g \left (x \right )+b \right )^{-n -1} f \left (x \right )^{-n +2} \left (\frac {d}{d x}g \left (x \right )\right )^{3}\right )^{n}-n^{-n} \left (\left (\frac {d}{d x}g \left (x \right )\right ) f \left (x \right )\right )^{-2 n +1} \left (\left (a g \left (x \right )+b \right )^{-n} f \left (x \right )^{-n +1} \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-n -1} \left (a n \left (a g \left (x \right )+b \right )^{-n -1} f \left (x \right )^{-n +2} \left (\frac {d}{d x}g \left (x \right )\right )^{3}\right )^{n}-\textit {\_a}^{n}}d \textit {\_a} \right )-\ln \left (a g \left (x \right )+b \right )\right ) f \left (x \right )}{a}\right \}\]