\[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, d^{\frac {1}{3}} \left (a b +\left (-a -b \right ) x +x^{2}\right )^{\frac {2}{3}}}{2 a^{2}-4 a x +2 x^{2}+d^{\frac {1}{3}} \left (a b +\left (-a -b \right ) x +x^{2}\right )^{\frac {2}{3}}}\right )}{2 d^{\frac {2}{3}}}+\frac {\ln \left (a^{2}-2 a x +x^{2}-d^{\frac {1}{3}} \left (a b +\left (-a -b \right ) x +x^{2}\right )^{\frac {2}{3}}\right )}{2 d^{\frac {2}{3}}}-\frac {\ln \left (a^{4}-4 a^{3} x +6 a^{2} x^{2}-4 x^{3} a +x^{4}+d^{\frac {2}{3}} \left (a b +\left (-a -b \right ) x +x^{2}\right )^{\frac {4}{3}}+\left (a b +\left (-a -b \right ) x +x^{2}\right )^{\frac {2}{3}} \left (a^{2} d^{\frac {1}{3}}-2 a \,d^{\frac {1}{3}} x +d^{\frac {1}{3}} x^{2}\right )\right )}{4 d^{\frac {2}{3}}} \]
command
Integrate[((-b + x)*(-(a*(a - 2*b)) - 2*b*x + x^2))/(((-a + x)*(-b + x))^(2/3)*(a^4 - b^2*d - 2*(2*a^3 - b*d)*x + (6*a^2 - d)*x^2 - 4*a*x^3 + x^4)),x]
Mathematica 13.1 output
\[ -\frac {\sqrt [3]{(a-x) (b-x)} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 (a-x)^{4/3}}{\sqrt [3]{d} (b-x)^{2/3}}}{\sqrt {3}}\right )-2 \log \left ((a-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x}\right )-2 \log \left ((a-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x}\right )+\log \left ((a-x)^{4/3}-\sqrt [6]{d} (a-x)^{2/3} \sqrt [3]{b-x}+\sqrt [3]{d} (b-x)^{2/3}\right )+\log \left ((a-x)^{4/3}+\sqrt [6]{d} (a-x)^{2/3} \sqrt [3]{b-x}+\sqrt [3]{d} (b-x)^{2/3}\right )\right )}{4 d^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x}} \]
Mathematica 12.3 output
\[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx \]________________________________________________________________________________________