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