4.19.36 $4x(a-x)(b-x)y^{\prime }(x{)}^2={(-2x(a+b)+ab+2x^2)}^2$

ODE
$$4x(a-x)(b-x)y^{\prime }(x{)}^2={(-2x(a+b)+ab+2x^2)}^2$$ ODE Classification

[_quadrature]

Book solution method
Change of variable

Mathematica ✓
cpu = 0.955217 (sec), leaf count = 297

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

Maple ✓
cpu = 0.069 (sec), leaf count = 83

$$\{y\left(x\right)=\int -\frac{2x^2+(-2a-2b)x+ab}{2}\frac{1}{\sqrt{x(b-x)(a-x)}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=\int \frac{2x^2+(-2a-2b)x+ab}{2}\frac{1}{\sqrt{x(b-x)(a-x)}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1}\}$$ 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] - ((I/3)*(-a + x)*(2*x*(-b + x) + (I*Sqrt[(-b + x)/(a - b)]*(2*(a
^2 - b^2)*EllipticE[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + b*(a + 2*b)*Elliptic
F[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]))/Sqrt[1 - a/x]))/Sqrt[-(x*(-a + x)*(-b 
+ x))]}, {y[x] -> C[1] + ((I/3)*(-a + x)*(2*x*(-b + x) + (I*Sqrt[(-b + x)/(a - b
)]*(2*(a^2 - b^2)*EllipticE[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + b*(a + 2*b)*
EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]))/Sqrt[1 - a/x]))/Sqrt[-(x*(-a +
 x)*(-b + x))]}}

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),'implicit')

Maple raw output

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