\[ \int \frac {(a-3 b+2 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-b+x) \sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (a b -a x -b x +x^{2}\right )^{\frac {3}{4}}}{b -x}-2 d^{\frac {1}{4}} \arctan \left (\frac {d^{\frac {1}{4}} \left (a b +\left (-a -b \right ) x +x^{2}\right )^{\frac {1}{4}}}{a -x}\right )+2 d^{\frac {1}{4}} \arctanh \left (\frac {d^{\frac {1}{4}} \left (a b +\left (-a -b \right ) x +x^{2}\right )^{\frac {1}{4}}}{a -x}\right ) \]
command
Integrate[((a - 3*b + 2*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/((-b + x)*((-a + x)*(-b + x))^(1/4)*(-a^3 + b*d - (-3*a^2 + d)*x - 3*a*x^2 + x^3)),x]
Mathematica 13.1 output
\[ \frac {4 (-a+x)}{\sqrt [4]{(-a+x) (-b+x)}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {a-x} \sqrt [4]{\frac {-b+x}{a-x}} \left (-\text {ArcTan}\left (\frac {-a+x+\sqrt {d} \sqrt {\frac {-b+x}{a-x}}}{\sqrt {2} \sqrt [4]{d} \sqrt {a-x} \sqrt [4]{\frac {-b+x}{a-x}}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {a-x} \sqrt [4]{\frac {b-x}{-a+x}}}{a-x+\sqrt {d} \sqrt {\frac {b-x}{-a+x}}}\right )\right )}{\sqrt [4]{(-a+x) (-b+x)}} \]
Mathematica 12.3 output
\[ \int \frac {(a-3 b+2 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-b+x) \sqrt [4]{(-a+x) (-b+x)} \left (-a^3+b d-\left (-3 a^2+d\right ) x-3 a x^2+x^3\right )} \, dx \]________________________________________________________________________________________