4.16.19 $f(x)(y(x)-a)(y(x)-b)+y^{\prime }(x{)}^2=0$

ODE
$$f(x)(y(x)-a)(y(x)-b)+y^{\prime }(x{)}^2=0$$ ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Binomial equation $(y^{\prime }{)}^m+F(x)G(y)=0$

Mathematica ✓
cpu = 0.239171 (sec), leaf count = 243

$$\{\{y(x)\to \frac{1}{4}(a^2(-e^{c_1+{\int }_1^x-i\sqrt{f(K[1])}dK[1]})+2a(1+b{e}^{c_1+{\int }_1^x-i\sqrt{f(K[1])}dK[1]})-e^{-c_1-{\int }_1^x-i\sqrt{f(K[1])}dK[1]}(-1+b{e}^{c_1+{\int }_1^x-i\sqrt{f(K[1])}dK[1]}){}^2)\},\{y(x)\to \frac{1}{4}(a^2(-e^{c_1+{\int }_1^{x}i\sqrt{f(K[2])}dK[2]})+2a(1+b{e}^{c_1+{\int }_1^{x}i\sqrt{f(K[2])}dK[2]})-e^{-c_1-{\int }_1^{x}i\sqrt{f(K[2])}dK[2]}(-1+b{e}^{c_1+{\int }_1^{x}i\sqrt{f(K[2])}dK[2]}){}^2)\}\}$$

Maple ✓
cpu = 0.396 (sec), leaf count = 212

$$\{1\sqrt{(y\left(x\right)-a)(y\left(x\right)-b)}\ln (-\frac{a}{2}-\frac{b}{2}+y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2+(-a-b)y\left(x\right)+ab})\frac{1}{\sqrt{y\left(x\right)-a}}\frac{1}{\sqrt{y\left(x\right)-b}}+{\int }^{x}1\sqrt{-f\left(\mathit{\_}\mathit{a}\right)(a-y\left(x\right))(b-y\left(x\right))}\frac{1}{\sqrt{y\left(x\right)-a}}\frac{1}{\sqrt{y\left(x\right)-b}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}=0,1\sqrt{(y\left(x\right)-a)(y\left(x\right)-b)}\ln (-\frac{a}{2}-\frac{b}{2}+y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2+(-a-b)y\left(x\right)+ab})\frac{1}{\sqrt{y\left(x\right)-a}}\frac{1}{\sqrt{y\left(x\right)-b}}+{\int }^x-1\sqrt{-f\left(\mathit{\_}\mathit{a}\right)(a-y\left(x\right))(b-y\left(x\right))}\frac{1}{\sqrt{y\left(x\right)-a}}\frac{1}{\sqrt{y\left(x\right)-b}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[f[x]*(-a + y[x])*(-b + y[x]) + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-(a^2*E^(C[1] + Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}])) - E^(-C
[1] - Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}])*(-1 + b*E^(C[1] + Integrate[(
-I)*Sqrt[f[K[1]]], {K[1], 1, x}]))^2 + 2*a*(1 + b*E^(C[1] + Integrate[(-I)*Sqrt[
f[K[1]]], {K[1], 1, x}])))/4}, {y[x] -> (-(a^2*E^(C[1] + Integrate[I*Sqrt[f[K[2]
]], {K[2], 1, x}])) - E^(-C[1] - Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}])*(-1 +
 b*E^(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}]))^2 + 2*a*(1 + b*E^(C[1] +
 Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}])))/4}}

Maple raw input

dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b) = 0, y(x),'implicit')

Maple raw output

((y(x)-a)*(y(x)-b))^(1/2)/(y(x)-a)^(1/2)/(y(x)-b)^(1/2)*ln(-1/2*a-1/2*b+y(x)+(y(
x)^2+(-a-b)*y(x)+a*b)^(1/2))+Intat(-(-f(_a)*(a-y(x))*(b-y(x)))^(1/2)/(y(x)-a)^(1
/2)/(y(x)-b)^(1/2),_a = x)+_C1 = 0, ((y(x)-a)*(y(x)-b))^(1/2)/(y(x)-a)^(1/2)/(y(
x)-b)^(1/2)*ln(-1/2*a-1/2*b+y(x)+(y(x)^2+(-a-b)*y(x)+a*b)^(1/2))+Intat((-f(_a)*(
a-y(x))*(b-y(x)))^(1/2)/(y(x)-a)^(1/2)/(y(x)-b)^(1/2),_a = x)+_C1 = 0