\[ -\frac {y(x)^2 f'(x)}{g(x)}+\frac {g'(x)}{f(x)}+y'(x)=0 \] ✓ Mathematica : cpu = 0.48339 (sec), leaf count = 160
\[\text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{(g(x)+f(x) K[2])^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2 f'(K[1])-g(K[1]) g'(K[1])\right )}{g(K[1]) (g(K[1])+f(K[1]) K[2])^3}-\frac {2 K[2] f'(K[1])}{g(K[1]) (g(K[1])+f(K[1]) K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2 f'(K[1])-g(K[1]) g'(K[1])}{f(K[1]) g(K[1]) (g(K[1])+f(K[1]) y(x))^2}dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.457 (sec), leaf count = 58
\[\left \{y \left (x \right ) = \frac {-c_{1} f \left (x \right ) g \left (x \right )-f \left (x \right ) g \left (x \right ) \left (\int \frac {\frac {d}{d x}f \left (x \right )}{f \left (x \right )^{2} g \left (x \right )}d x \right )-1}{\left (c_{1}+\int \frac {\frac {d}{d x}f \left (x \right )}{f \left (x \right )^{2} g \left (x \right )}d x \right ) f \left (x \right )^{2}}\right \}\]
\begin {align} -\frac {f^{\prime }}{g}y^{2}+\frac {g^{\prime }}{f}+y^{\prime } & =0\nonumber \\ y^{\prime } & =-\frac {g^{\prime }}{f}+\frac {f^{\prime }}{g}y^{2}\nonumber \\ & =P\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}\tag {1} \end {align}
This is Ricatti first order non-linear ODE. \(P\left ( x\right ) =-\frac {g^{\prime }}{f},Q\left ( x\right ) =0,R\left ( x\right ) =\frac {f^{\prime }}{g}\).
To do.