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