##### 4.16.22 $$f(x) (y(x)-a)^2 (y(x)-b) (y(x)-c)+y'(x)^2=0$$

ODE
$f(x) (y(x)-a)^2 (y(x)-b) (y(x)-c)+y'(x)^2=0$ 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 = 0.375737 (sec), leaf count = 251

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

Maple
cpu = 0.321 (sec), leaf count = 378

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

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

Mathematica raw output

{{y[x] -> (b*(-a + c) + (a - b)*c*Tanh[(Sqrt[a - b]*Sqrt[a - c]*(C[1] + Inactive
[Integrate][(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]))/2]^2)/(-a + c + (a - b)*Tanh[(Sq
rt[a - b]*Sqrt[a - c]*(C[1] + Inactive[Integrate][(-I)*Sqrt[f[K[1]]], {K[1], 1,
x}]))/2]^2)}, {y[x] -> (b*(-a + c) + (a - b)*c*Tanh[(Sqrt[a - b]*Sqrt[a - c]*(C[
1] + Inactive[Integrate][I*Sqrt[f[K[2]]], {K[2], 1, x}]))/2]^2)/(-a + c + (a - b
)*Tanh[(Sqrt[a - b]*Sqrt[a - c]*(C[1] + Inactive[Integrate][I*Sqrt[f[K[2]]], {K[
2], 1, x}]))/2]^2)}}

Maple raw input

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

Maple raw output

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