\[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-a \,b^{2}+\left (2 a b +b^{2}\right ) x +\left (-a -2 b \right ) x^{2}+x^{3}\right )^{\frac {2}{3}}}{-2 a \,d^{\frac {1}{3}}+2 d^{\frac {1}{3}} x +\left (-a \,b^{2}+\left (2 a b +b^{2}\right ) x +\left (-a -2 b \right ) x^{2}+x^{3}\right )^{\frac {2}{3}}}\right )}{d^{\frac {1}{3}}}+\frac {\ln \left (a \sqrt {d}-x \sqrt {d}+d^{\frac {1}{6}} \left (-a \,b^{2}+\left (2 a b +b^{2}\right ) x +\left (-a -2 b \right ) x^{2}+x^{3}\right )^{\frac {2}{3}}\right )}{d^{\frac {1}{3}}}-\frac {\ln \left (a^{2} d -2 a d x +d \,x^{2}+\left (-a \,d^{\frac {2}{3}}+d^{\frac {2}{3}} x \right ) \left (-a \,b^{2}+\left (2 a b +b^{2}\right ) x +\left (-a -2 b \right ) x^{2}+x^{3}\right )^{\frac {2}{3}}+d^{\frac {1}{3}} \left (-a \,b^{2}+\left (2 a b +b^{2}\right ) x +\left (-a -2 b \right ) x^{2}+x^{3}\right )^{\frac {4}{3}}\right )}{2 d^{\frac {1}{3}}} \]
command
Integrate[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x)^2)^(2/3)*(b^4 + a*d - (4*b^3 + d)*x + 6*b^2*x^2 - 4*b*x^3 + x^4)),x]
Mathematica 13.1 output
\[ -\frac {(a-x)^{2/3} (b-x)^{4/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a-x}}{(b-x)^{4/3}}}{\sqrt {3}}\right )+\log \left (\frac {(a-b)^2 \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} (b-x)^{4/3}+(b-x)^{8/3}\right )}{(b-x)^{8/3}}\right )-2 \log \left (\frac {(a-b) \left (\sqrt [3]{d} \sqrt [3]{a-x}+(b-x)^{4/3}\right )}{(b-x)^{4/3}}\right )\right )}{2 \sqrt [3]{d} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx \]________________________________________________________________________________________