\[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{\left (-a +x \right ) \left (-b +x \right )}-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {d}\, \sqrt {a b x +\left (-a -b \right ) x^{2}+x^{3}}}\right ) \]
command
Int[(-(a*b*x) + x^3)/((-a + x)*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a*b*d - (1 + a*d + b*d)*x + d*x^2)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {output too large to display} \]
Rubi 4.16.1 under Mathematica 13.3.1 output \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx \]__________________________________________