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