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