\[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {a b x +\left (-a -b \right ) x^{2}+x^{3}}}{x}-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {d}\, x}{\sqrt {a b x +\left (-a -b \right ) x^{2}+x^{3}}}\right ) \]
command
Int[((-a + x)*(-b + x)*(-(a*b) + x^2))/(x*Sqrt[x*(-a + x)*(-b + x)]*(a*b - (a + b + d)*x + x^2)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {2 \left (x^2-(a+b) x+a b\right )}{\sqrt {(a-x) (b-x) x}}-\frac {2 \sqrt {d} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {x^2-(a+b) x+a b}}\right ) \sqrt {x^2-(a+b) x+a b}}{\sqrt {(a-x) (b-x) x}}+\frac {d \sqrt {x} \left (x+\sqrt {a} \sqrt {b}\right ) \sqrt {\frac {x^2-(a+b) x+a b}{\left (x+\sqrt {a} \sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {(a-x) (b-x) x}}-\frac {d \left (a+b+d-\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {x} \left (x+\sqrt {a} \sqrt {b}\right ) \sqrt {\frac {x^2-(a+b) x+a b}{\left (x+\sqrt {a} \sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{\sqrt [4]{a} \sqrt [4]{b} \left (a+2 \sqrt {b} \sqrt {a}+b+d-\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}}-\frac {d \left (a+b+d+\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {x} \left (x+\sqrt {a} \sqrt {b}\right ) \sqrt {\frac {x^2-(a+b) x+a b}{\left (x+\sqrt {a} \sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{\sqrt [4]{a} \sqrt [4]{b} \left (a+2 \sqrt {b} \sqrt {a}+b+d+\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}}+\frac {d \left (a-2 \sqrt {b} \sqrt {a}+b+d-\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {x} \left (x+\sqrt {a} \sqrt {b}\right ) \sqrt {\frac {x^2-(a+b) x+a b}{\left (x+\sqrt {a} \sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {a+2 \sqrt {b} \sqrt {a}+b+d}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (a+2 \sqrt {b} \sqrt {a}+b+d-\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}}+\frac {d \left (a-2 \sqrt {b} \sqrt {a}+b+d+\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {x} \left (x+\sqrt {a} \sqrt {b}\right ) \sqrt {\frac {x^2-(a+b) x+a b}{\left (x+\sqrt {a} \sqrt {b}\right )^2}} \operatorname {EllipticPi}\left (\frac {a+2 \sqrt {b} \sqrt {a}+b+d}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (a+2 \sqrt {b} \sqrt {a}+b+d+\sqrt {a^2-2 (b-d) a+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx \]________________________________________________________________________________________