4.16.16 $y^{\prime }(x{)}^2=(y(x)-a)(y(x)-b)(y(x)-c)$

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

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.602337 (sec), leaf count = 173

$$\{\{y(x)\to \text{ns}\left(\frac{1}{2}\sqrt{a-b}(c_1-ix)|\frac{a-c}{a-b}\right){}^2(a\text{sn}\left(\frac{1}{2}\sqrt{a-b}(c_1-ix)|\frac{a-c}{a-b}\right){}^2-a+b)\},\{y(x)\to \text{ns}\left(\frac{1}{2}\sqrt{a-b}(ix+c_1)|\frac{a-c}{a-b}\right){}^2(a\text{sn}\left(\frac{1}{2}\sqrt{a-b}(ix+c_1)|\frac{a-c}{a-b}\right){}^2-a+b)\}\}$$

Maple ✓
cpu = 0.159 (sec), leaf count = 90

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

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

Mathematica raw output

{{y[x] -> JacobiNS[(Sqrt[a - b]*((-I)*x + C[1]))/2, (a - c)/(a - b)]^2*(-a + b +
 a*JacobiSN[(Sqrt[a - b]*((-I)*x + C[1]))/2, (a - c)/(a - b)]^2)}, {y[x] -> Jaco
biNS[(Sqrt[a - b]*(I*x + C[1]))/2, (a - c)/(a - b)]^2*(-a + b + a*JacobiSN[(Sqrt
[a - b]*(I*x + C[1]))/2, (a - c)/(a - b)]^2)}}

Maple raw input

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

Maple raw output

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