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