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