##### 4.19.36 $$4 x (a-x) (b-x) y'(x)^2=\left (-2 x (a+b)+a b+2 x^2\right )^2$$

ODE
$4 x (a-x) (b-x) y'(x)^2=\left (-2 x (a+b)+a b+2 x^2\right )^2$ ODE Classiﬁcation

Book solution method
Change of variable

Mathematica
cpu = 1.32631 (sec), leaf count = 375

$\left \{\left \{y(x)\to c_1-\frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}}\right \},\left \{y(x)\to \frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}}+c_1\right \}\right \}$

Maple
cpu = 0.072 (sec), leaf count = 81

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

DSolve[4*(a - x)*(b - x)*x*y'[x]^2 == (a*b - 2*(a + b)*x + 2*x^2)^2,y[x],x]

Mathematica raw output

{{y[x] -> C[1] - ((a - x)*((2*I)*(b - x)*x*Sqrt[-1 + x/a] + 2*(a^2 - b^2)*Sqrt[x
/a]*Sqrt[(-b + x)/(a - b)]*EllipticE[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + b*(
a + 2*b)*Sqrt[x/a]*Sqrt[(-b + x)/(a - b)]*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a
/(a - b)]))/(3*Sqrt[(a - x)*x*(-b + x)]*Sqrt[-1 + x/a])}, {y[x] -> C[1] + ((a -
x)*((2*I)*(b - x)*x*Sqrt[-1 + x/a] + 2*(a^2 - b^2)*Sqrt[x/a]*Sqrt[(-b + x)/(a -
b)]*EllipticE[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + b*(a + 2*b)*Sqrt[x/a]*Sqrt
[(-b + x)/(a - b)]*EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]))/(3*Sqrt[(a
- x)*x*(-b + x)]*Sqrt[-1 + x/a])}}

Maple raw input

dsolve(4*x*(a-x)*(b-x)*diff(y(x),x)^2 = (a*b-2*x*(a+b)+2*x^2)^2, y(x))

Maple raw output

[y(x) = Int(-1/2/(x*(a-x)*(b-x))^(1/2)*(a*b-2*a*x-2*b*x+2*x^2),x)+_C1, y(x) = In
t(1/2/(x*(a-x)*(b-x))^(1/2)*(a*b-2*a*x-2*b*x+2*x^2),x)+_C1]