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