\[ f(x) \left (2 a y(x)+b+y(x)^2\right )+y'(x)=0 \] ✓ Mathematica : cpu = 0.123954 (sec), leaf count = 61
\[\left \{\left \{y(x)\to -a+\sqrt {b-a^2} \tan \left (\sqrt {b-a^2} \int _1^x-f(K[1])dK[1]+c_1 \sqrt {b-a^2}\right )\right \}\right \}\] ✓ Maple : cpu = 0.046 (sec), leaf count = 35
\[y \relax (x ) = \tanh \left (\sqrt {a^{2}-b}\, \left (c_{1}+\int f \relax (x )d x \right )\right ) \sqrt {a^{2}-b}-a\]
Hand solution
\begin {align} y^{\prime }\relax (x) +f\relax (x) \left (2ay\relax (x) +b+y^{2}\relax (x) \right ) & =0\nonumber \\ y^{\prime }\relax (x) & =-2af\relax (x) y\relax (x) -bf\relax (x) -f\relax (x) y^{2}\relax (x) \nonumber \\ & =P\relax (x) +Q\relax (x) y+R\relax (x) y^{2} \tag {1} \end {align}
This is Riccati first order non-linear ODE. \(P\relax (x) =-bf\left ( x\right ) ,Q\relax (x) =-2af\relax (x) ,R\relax (x) =-f\relax (x) \).
Let \[ y\relax (x) =-\frac {u^{\prime }\relax (x) }{u\relax (x) R\relax (x) }=\frac {u^{\prime }\relax (x) }{u\relax (x) f\relax (x) }\]
Hence
\[ y^{\prime }\relax (x) =\frac {u^{\prime \prime }\relax (x) }{u\relax (x) f\relax (x) }-\frac {\left (u^{\prime }\left ( x\right ) \right ) ^{2}}{u^{2}\relax (x) f\relax (x) }-\frac {u^{\prime }\relax (x) f^{\prime }\relax (x) }{u\left ( x\right ) f^{2}\relax (x) }\]
Equating this to RHS of (1) gives
\begin {align*} \frac {u^{\prime \prime }\relax (x) }{u\relax (x) f\left ( x\right ) }-\frac {\left (u^{\prime }\relax (x) \right ) ^{2}}{u^{2}\relax (x) f\relax (x) }-\frac {u^{\prime }\relax (x) f^{\prime }\relax (x) }{u\relax (x) f^{2}\relax (x) } & =-2af\relax (x) y\relax (x) -bf\relax (x) -f\left ( x\right ) y^{2}\relax (x) \\ & =-2af\relax (x) \left [ \frac {u^{\prime }\relax (x) }{u\left ( x\right ) f\relax (x) }\right ] -bf\relax (x) -f\left ( x\right ) \left [ \frac {u^{\prime }\relax (x) }{u\relax (x) f\relax (x) }\right ] ^{2}\\ & =-2a\frac {u^{\prime }\relax (x) }{u\relax (x) }-bf\left ( x\right ) -\frac {u^{\prime }\relax (x) ^{2}}{u^{2}\relax (x) f\relax (x) } \end {align*}
Simplifying
\begin {align*} u^{\prime \prime }\relax (x) -\frac {\left (u^{\prime }\relax (x) \right ) ^{2}}{u\relax (x) }-\frac {u^{\prime }\relax (x) f^{\prime }\relax (x) }{f\relax (x) } & =-2au^{\prime }\left ( x\right ) f\relax (x) -u\relax (x) bf^{2}\relax (x) -\frac {u^{\prime }\relax (x) ^{2}}{u\relax (x) }\\ u^{\prime \prime }\relax (x) -\frac {u^{\prime }\relax (x) f^{\prime }\relax (x) }{f\relax (x) } & =-2au^{\prime }\left ( x\right ) f\relax (x) -u\relax (x) bf^{2}\relax (x) \\ u^{\prime \prime }\relax (x) +u^{\prime }\relax (x) \left ( -\frac {f^{\prime }\relax (x) }{f\relax (x) }+2af\relax (x) \right ) +u\relax (x) bf^{2}\relax (x) & =0 \end {align*}
Second order ODE with variable coefficients. Since coefficients are variables and not constants, a power series method is the standard way to continue. When I tried solving this now pretending the coefficients are constants in time, using the standard auxiliary equation method, the solution did verify OK. I need to look more into this. For now, this is solved using standard method for solving second order ODE with constant coefficients.
\[ u\relax (x) =C_{1}\exp \left (\frac {\int f\relax (x) \sqrt {-b}dx\left (\sqrt {\frac {b-a^{2}}{b}}b+a\sqrt {-b}\right ) }{b}\right ) +C_{2}\exp \left (\frac {\int f\relax (x) \sqrt {-b}dx\left ( -\sqrt {\frac {b-a^{2}}{b}}b+a\sqrt {-b}\right ) }{b}\right ) \]
Hence
\begin {align*} u^{\prime }\relax (x) & ={\frac {C_{1}\,f\relax (x) \sqrt {-b}}{b}\left (\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) e{^{{\frac {\int \!f\relax (x) \sqrt {-b}\,\mathrm {d}x}{b}\left (\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) }}}}\\ & +{\frac {C_{2}\,f\relax (x) \sqrt {-b}}{b}\left (-\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) e{^{{\frac {\int \!f\relax (x) \sqrt {-b}\,\mathrm {d}x}{b}\left (-\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) }}}} \end {align*}
Therefore
\begin {align*} y & =\frac {u^{\prime }\relax (x) }{u\relax (x) f\left ( x\right ) }\\ & =\frac {{\frac {C_{1}\,f\relax (x) \sqrt {-b}}{b}\left (\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) e{^{{\frac {\int \!f\left ( x\right ) \sqrt {-b}\,\mathrm {d}x}{b}\left (\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) }}+}\frac {C_{2}\,f\relax (x) \sqrt {-b}}{b}\left ( -\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) e{^{{\frac {\int \!f\left ( x\right ) \sqrt {-b}\,\mathrm {d}x}{b}\left (-\sqrt {{\frac {-{a}^{2}+b}{b}}}b+\sqrt {-b}a\right ) }}}}}{f\relax (x) \left [ C_{1}\exp \left ( \frac {\int f\relax (x) \sqrt {-b}dx\left (\sqrt {\frac {b-a^{2}}{b}}b+a\sqrt {-b}\right ) }{b}\right ) +C_{2}\exp \left (\frac {\int f\left ( x\right ) \sqrt {-b}dx\left (-\sqrt {\frac {b-a^{2}}{b}}b+a\sqrt {-b}\right ) }{b}\right ) \right ] } \end {align*}
Verification