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