\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {a b x +\left (-a -b \right ) x^{2}+x^{3}}}{a -x}+2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {a b x +\left (-a -b \right ) x^{2}+x^{3}}}{\sqrt {d}\, \left (a -x \right )}\right ) \]
command
Int[(x*(-b + x)*(a*b - 2*a*x + x^2))/((-a + x)*Sqrt[x*(-a + x)*(-b + x)]*(a*d + (-b - d)*x + x^2)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
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Rubi 4.16.1 under Mathematica 13.3.1 output \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx \]__________________________________________