##### 4.16.19 $$f(x) (y(x)-a) (y(x)-b)+y'(x)^2=0$$

ODE
$f(x) (y(x)-a) (y(x)-b)+y'(x)^2=0$ ODE Classiﬁcation

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

Book solution method
Binomial equation $$(y')^m + F(x) G(y)=0$$

Mathematica
cpu = 0.353267 (sec), leaf count = 175

$\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-b} \sqrt {\frac {\text {\#1}-b}{a-b}} \sin ^{-1}\left (\frac {\sqrt {a-\text {\#1}}}{\sqrt {a-b}}\right )}{\sqrt {b-\text {\#1}}}\& \right ]\left [\int _1^x-i \sqrt {f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-b} \sqrt {\frac {\text {\#1}-b}{a-b}} \sin ^{-1}\left (\frac {\sqrt {a-\text {\#1}}}{\sqrt {a-b}}\right )}{\sqrt {b-\text {\#1}}}\& \right ]\left [\int _1^xi \sqrt {f(K[3])}dK[3]+c_1\right ]\right \}\right \}$

Maple
cpu = 0.348 (sec), leaf count = 212

$\left [\frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \ln \left (-\frac {a}{2}-\frac {b}{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+\left (-a -b \right ) y \left (x \right )+a b}\right )}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}+\int _{}^{x}-\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}d \textit {\_a} +\textit {\_C1} = 0, \frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \ln \left (-\frac {a}{2}-\frac {b}{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+\left (-a -b \right ) y \left (x \right )+a b}\right )}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}+\int _{}^{x}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}d \textit {\_a} +\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> InverseFunction[(-2*Sqrt[a - b]*ArcSin[Sqrt[a - #1]/Sqrt[a - b]]*Sqrt[
(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][(-I)*Sqrt[f[K[2]
]], {K[2], 1, x}]]}, {y[x] -> InverseFunction[(-2*Sqrt[a - b]*ArcSin[Sqrt[a - #1
]/Sqrt[a - b]]*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integra
te][I*Sqrt[f[K[3]]], {K[3], 1, x}]]}}

Maple raw input

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

Maple raw output

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