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