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

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

odeadvisor timed out

Book solution method
Binomial equation $$(y')^m + F(x) G(y)=0$$

Mathematica
cpu = 2.41114 (sec), leaf count = 415

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

Maple
cpu = 0.317 (sec), leaf count = 190

$\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (-\textit {\_a} +\mathit {a4} \right ) \left (-\textit {\_a} +\mathit {a3} \right ) \left (-\textit {\_a} +\mathit {a2} \right ) \left (-\textit {\_a} +\mathit {a1} \right )}}d \textit {\_a} +\int _{}^{x}-\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \left (x \right )+\mathit {a4} \right ) \left (-y \left (x \right )+\mathit {a3} \right ) \left (-y \left (x \right )+\mathit {a2} \right ) \left (-y \left (x \right )+\mathit {a1} \right )}}{\sqrt {\left (-y \left (x \right )+\mathit {a4} \right ) \left (-y \left (x \right )+\mathit {a3} \right ) \left (-y \left (x \right )+\mathit {a2} \right ) \left (-y \left (x \right )+\mathit {a1} \right )}}d \textit {\_a} +\textit {\_C1} = 0, \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (-\textit {\_a} +\mathit {a4} \right ) \left (-\textit {\_a} +\mathit {a3} \right ) \left (-\textit {\_a} +\mathit {a2} \right ) \left (-\textit {\_a} +\mathit {a1} \right )}}d \textit {\_a} +\int _{}^{x}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \left (x \right )+\mathit {a4} \right ) \left (-y \left (x \right )+\mathit {a3} \right ) \left (-y \left (x \right )+\mathit {a2} \right ) \left (-y \left (x \right )+\mathit {a1} \right )}}{\sqrt {\left (-y \left (x \right )+\mathit {a4} \right ) \left (-y \left (x \right )+\mathit {a3} \right ) \left (-y \left (x \right )+\mathit {a2} \right ) \left (-y \left (x \right )+\mathit {a1} \right )}}d \textit {\_a} +\textit {\_C1} = 0\right ]$ 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] + Inactive[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[a
2 - #1]*Sqrt[a4 - #1]*Sqrt[((a1 - a2)*(-a3 + #1))/((-a2 + a3)*(-a1 + #1))])/((a2
 - a4)*Sqrt[a1 - #1]*Sqrt[a3 - #1]*Sqrt[((a1 - a2)*(a1 - a4)*(a2 - #1)*(-a4 + #1
))/((a2 - a4)^2*(a1 - #1)^2)]) & ][C[1] + Inactive[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))

Maple raw output

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