##### 4.16.24 $$y'(x)^2=f(x)^2 (y(x)-a) (y(x)-b) (y(x)-c)^2$$

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

[_separable]

Book solution method
Change of variable

Mathematica
cpu = 0.35263 (sec), leaf count = 223

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

Maple
cpu = 0.749 (sec), leaf count = 821

$\left [y \left (x \right ) = \frac {a^{2} c -2 b c a +4 a b \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}-2 a c \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}+b^{2} c -2 b c \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}+c \,{\mathrm e}^{2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+2 \textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}}{a^{2}-2 a b +2 a \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}+b^{2}+2 b \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}-4 \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}} c +{\mathrm e}^{2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}+2 \textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}}, y \left (x \right ) = \frac {a^{2} c -2 b c a +4 a b \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}-2 a c \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}+b^{2} c -2 b c \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}+c \,{\mathrm e}^{-2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-2 \textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}}{a^{2}-2 a b +2 a \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}+b^{2}+2 b \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}-4 \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-\textit {\_C1} \sqrt {a b -c a -c b +c^{2}}} c +{\mathrm e}^{-2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -c a -c b +c^{2}}-2 \textit {\_C1} \sqrt {a b -c a -c b +c^{2}}}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (a*(-b + c) + b*(a - c)*Tanh[(Sqrt[-a + c]*Sqrt[-b + c]*(C[1] + Inacti
ve[Integrate][-f[K[1]], {K[1], 1, x}]))/2]^2)/(-b + c + (a - c)*Tanh[(Sqrt[-a +
c]*Sqrt[-b + c]*(C[1] + Inactive[Integrate][-f[K[1]], {K[1], 1, x}]))/2]^2)}, {y
[x] -> (a*(-b + c) + b*(a - c)*Tanh[(Sqrt[-a + c]*Sqrt[-b + c]*(C[1] + Inactive[
Integrate][f[K[2]], {K[2], 1, x}]))/2]^2)/(-b + c + (a - c)*Tanh[(Sqrt[-a + c]*S
qrt[-b + c]*(C[1] + Inactive[Integrate][f[K[2]], {K[2], 1, x}]))/2]^2)}}

Maple raw input

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

Maple raw output

[y(x) = (a^2*c-2*b*c*a+4*a*b*exp(Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+_C1*(a*b-a*
c-b*c+c^2)^(1/2))-2*a*c*exp(Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+_C1*(a*b-a*c-b*c
+c^2)^(1/2))+b^2*c-2*b*c*exp(Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+_C1*(a*b-a*c-b*
c+c^2)^(1/2))+c*exp(2*Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+2*_C1*(a*b-a*c-b*c+c^2
)^(1/2)))/(a^2-2*a*b+2*a*exp(Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+_C1*(a*b-a*c-b*
c+c^2)^(1/2))+b^2+2*b*exp(Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+_C1*(a*b-a*c-b*c+c
^2)^(1/2))-4*exp(Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+_C1*(a*b-a*c-b*c+c^2)^(1/2)
)*c+exp(2*Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)+2*_C1*(a*b-a*c-b*c+c^2)^(1/2))), y
(x) = (a^2*c-2*b*c*a+4*a*b*exp(-Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-_C1*(a*b-a*c
-b*c+c^2)^(1/2))-2*a*c*exp(-Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-_C1*(a*b-a*c-b*c
+c^2)^(1/2))+b^2*c-2*b*c*exp(-Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-_C1*(a*b-a*c-b
*c+c^2)^(1/2))+c*exp(-2*Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-2*_C1*(a*b-a*c-b*c+c
^2)^(1/2)))/(a^2-2*a*b+2*a*exp(-Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-_C1*(a*b-a*c
-b*c+c^2)^(1/2))+b^2+2*b*exp(-Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-_C1*(a*b-a*c-b
*c+c^2)^(1/2))-4*exp(-Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-_C1*(a*b-a*c-b*c+c^2)^
(1/2))*c+exp(-2*Int(f(x),x)*(a*b-a*c-b*c+c^2)^(1/2)-2*_C1*(a*b-a*c-b*c+c^2)^(1/2
)))]