##### 4.16.21 $$f(x) (y(x)-a) (y(x)-b) (y(x)-c)+y'(x)^2=0$$

ODE
$f(x) (y(x)-a) (y(x)-b) (y(x)-c)+y'(x)^2=0$ 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 = 0.495198 (sec), leaf count = 213

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

Maple
cpu = 0.372 (sec), leaf count = 158

$\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right )}}d \textit {\_a} +\int _{}^{x}-\frac {\sqrt {f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (a -y \left (x \right )\right )}}{\sqrt {-\left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (a -y \left (x \right )\right )}}d \textit {\_a} +\textit {\_C1} = 0, \int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right )}}d \textit {\_a} +\int _{}^{x}\frac {\sqrt {f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (a -y \left (x \right )\right )}}{\sqrt {-\left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (a -y \left (x \right )\right )}}d \textit {\_a} +\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> JacobiNS[(Sqrt[a - b]*(C[1] + Inactive[Integrate][-Sqrt[f[K[1]]], {K[1
], 1, x}]))/2, (a - c)/(a - b)]^2*(-a + b + a*JacobiSN[(Sqrt[a - b]*(C[1] + Inac
tive[Integrate][-Sqrt[f[K[1]]], {K[1], 1, x}]))/2, (a - c)/(a - b)]^2)}, {y[x] -
> JacobiNS[(Sqrt[a - b]*(C[1] + Inactive[Integrate][Sqrt[f[K[2]]], {K[2], 1, x}]
))/2, (a - c)/(a - b)]^2*(-a + b + a*JacobiSN[(Sqrt[a - b]*(C[1] + Inactive[Inte
grate][Sqrt[f[K[2]]], {K[2], 1, x}]))/2, (a - c)/(a - b)]^2)}}

Maple raw input

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

Maple raw output

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