4.16.23 $f(x)(y(x)-\text{a1})(y(x)-\text{a2})(y(x)-\text{a3})(y(x)-\text{a4})+y^{\prime }(x{)}^2=0$

ODE
$$f(x)(y(x)-\text{a1})(y(x)-\text{a2})(y(x)-\text{a3})(y(x)-\text{a4})+y^{\prime }(x{)}^2=0$$ ODE Classification

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

Book solution method
Binomial equation $(y^{\prime }{)}^m+F(x)G(y)=0$

Mathematica ✓
cpu = 2.85498 (sec), leaf count = 413

$$\{\{y(x)\to \text{InverseFunction}\left[\frac{2\sqrt{\text{a2}-\mathrm{\#}\text{1}}\sqrt{\text{a4}-\mathrm{\#}\text{1}}\sqrt{\frac{(\mathrm{\#}\text{1}-\text{a3})(\text{a1}-\text{a2})}{(\mathrm{\#}\text{1}-\text{a1})(\text{a3}-\text{a2})}}F({\sin }^{-1}\left(\sqrt{\frac{(\text{a1}-\text{a4})(\mathrm{\#}\text{1}-\text{a2})}{(\text{a2}-\text{a4})(\mathrm{\#}\text{1}-\text{a1})}}\right)|\frac{(\text{a1}-\text{a3})(\text{a2}-\text{a4})}{(\text{a2}-\text{a3})(\text{a1}-\text{a4})})}{\sqrt{\text{a1}-\mathrm{\#}\text{1}}\sqrt{\text{a3}-\mathrm{\#}\text{1}}(\text{a2}-\text{a4})\sqrt{\frac{(\text{a2}-\mathrm{\#}\text{1})(\mathrm{\#}\text{1}-\text{a4})(\text{a1}-\text{a2})(\text{a1}-\text{a4})}{(\text{a1}-\mathrm{\#}\text{1}{)}^2(\text{a2}-\text{a4}{)}^2}}}\mathrm{\&}\right][c_1+{\int }_1^x-i\sqrt{f(K[1])}dK[1]]\},\{y(x)\to \text{InverseFunction}\left[\frac{2\sqrt{\text{a2}-\mathrm{\#}\text{1}}\sqrt{\text{a4}-\mathrm{\#}\text{1}}\sqrt{\frac{(\mathrm{\#}\text{1}-\text{a3})(\text{a1}-\text{a2})}{(\mathrm{\#}\text{1}-\text{a1})(\text{a3}-\text{a2})}}F({\sin }^{-1}\left(\sqrt{\frac{(\text{a1}-\text{a4})(\mathrm{\#}\text{1}-\text{a2})}{(\text{a2}-\text{a4})(\mathrm{\#}\text{1}-\text{a1})}}\right)|\frac{(\text{a1}-\text{a3})(\text{a2}-\text{a4})}{(\text{a2}-\text{a3})(\text{a1}-\text{a4})})}{\sqrt{\text{a1}-\mathrm{\#}\text{1}}\sqrt{\text{a3}-\mathrm{\#}\text{1}}(\text{a2}-\text{a4})\sqrt{\frac{(\text{a2}-\mathrm{\#}\text{1})(\mathrm{\#}\text{1}-\text{a4})(\text{a1}-\text{a2})(\text{a1}-\text{a4})}{(\text{a1}-\mathrm{\#}\text{1}{)}^2(\text{a2}-\text{a4}{)}^2}}}\mathrm{\&}\right][c_1+{\int }_1^{x}i\sqrt{f(K[2])}dK[2]]\}\}$$

Maple ✓
cpu = 0.242 (sec), leaf count = 190

$$\{{\int }^{y\left(x\right)}\frac{1}{\sqrt{(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{4})(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{3})(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{2})(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{1})}}d\mathit{\_}\mathit{a}+{\int }^{x}1\sqrt{-f\left(\mathit{\_}\mathit{a}\right)(\mathit{a}\mathit{4}-y\left(x\right))(\mathit{a}\mathit{3}-y\left(x\right))(\mathit{a}\mathit{2}-y\left(x\right))(\mathit{a}\mathit{1}-y\left(x\right))}\frac{1}{\sqrt{(\mathit{a}\mathit{4}-y\left(x\right))(\mathit{a}\mathit{3}-y\left(x\right))(\mathit{a}\mathit{2}-y\left(x\right))(\mathit{a}\mathit{1}-y\left(x\right))}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}=0,{\int }^{y\left(x\right)}\frac{1}{\sqrt{(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{4})(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{3})(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{2})(-\mathit{\_}\mathit{a}+\mathit{a}\mathit{1})}}d\mathit{\_}\mathit{a}+{\int }^x-1\sqrt{-f\left(\mathit{\_}\mathit{a}\right)(\mathit{a}\mathit{4}-y\left(x\right))(\mathit{a}\mathit{3}-y\left(x\right))(\mathit{a}\mathit{2}-y\left(x\right))(\mathit{a}\mathit{1}-y\left(x\right))}\frac{1}{\sqrt{(\mathit{a}\mathit{4}-y\left(x\right))(\mathit{a}\mathit{3}-y\left(x\right))(\mathit{a}\mathit{2}-y\left(x\right))(\mathit{a}\mathit{1}-y\left(x\right))}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[f[x]*(-a1 + y[x])*(-a2 + y[x])*(-a3 + y[x])*(-a4 + y[x]) + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(2*EllipticF[ArcSin[Sqrt[((a1 - a4)*(-a2 + #1))/((a2 -
 a4)*(-a1 + #1))]], ((a1 - a3)*(a2 - a4))/((a2 - a3)*(a1 - a4))]*Sqrt[a2 - #1]*S
qrt[a4 - #1]*Sqrt[((a1 - a2)*(-a3 + #1))/((-a2 + a3)*(-a1 + #1))])/((a2 - a4)*Sq
rt[a1 - #1]*Sqrt[a3 - #1]*Sqrt[((a1 - a2)*(a1 - a4)*(a2 - #1)*(-a4 + #1))/((a2 -
 a4)^2*(a1 - #1)^2)]) & ][C[1] + Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]]}, 
{y[x] -> InverseFunction[(2*EllipticF[ArcSin[Sqrt[((a1 - a4)*(-a2 + #1))/((a2 - 
a4)*(-a1 + #1))]], ((a1 - a3)*(a2 - a4))/((a2 - a3)*(a1 - a4))]*Sqrt[a2 - #1]*Sq
rt[a4 - #1]*Sqrt[((a1 - a2)*(-a3 + #1))/((-a2 + a3)*(-a1 + #1))])/((a2 - a4)*Sqr
t[a1 - #1]*Sqrt[a3 - #1]*Sqrt[((a1 - a2)*(a1 - a4)*(a2 - #1)*(-a4 + #1))/((a2 - 
a4)^2*(a1 - #1)^2)]) & ][C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}]]}}

Maple raw input

dsolve(diff(y(x),x)^2+f(x)*(y(x)-a1)*(y(x)-a2)*(y(x)-a3)*(y(x)-a4) = 0, y(x),'implicit')

Maple raw output

Intat(1/((-_a+a4)*(-_a+a3)*(-_a+a2)*(-_a+a1))^(1/2),_a = y(x))+Intat(-(-f(_a)*(a
4-y(x))*(a3-y(x))*(a2-y(x))*(a1-y(x)))^(1/2)/((a4-y(x))*(a3-y(x))*(a2-y(x))*(a1-
y(x)))^(1/2),_a = x)+_C1 = 0, Intat(1/((-_a+a4)*(-_a+a3)*(-_a+a2)*(-_a+a1))^(1/2
),_a = y(x))+Intat((-f(_a)*(a4-y(x))*(a3-y(x))*(a2-y(x))*(a1-y(x)))^(1/2)/((a4-y
(x))*(a3-y(x))*(a2-y(x))*(a1-y(x)))^(1/2),_a = x)+_C1 = 0