\[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx \]
Optimal antiderivative \[ -\sqrt {3}\, \arctan \! \left (\frac {612314840 \sqrt {3}\, \left (x^{3}-x \right )^{\frac {1}{3}} \left (-1+x \right )+\sqrt {3}\, \left (1609127381 x^{2}+1235276981 x +124616800\right )+2605939922 \sqrt {3}\, \left (x^{3}-x \right )^{\frac {2}{3}}}{2990437623 x^{2}+3108349623 x -39304000}\right )-\frac {\ln \! \left (\frac {3 \left (-1+x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}+3 x -3 \left (x^{3}-x \right )^{\frac {2}{3}}-1}{-1+3 x}\right )}{2} \]
command
Int[((-1 + x)*(1 + 3*x))/((-1 + 3*x)*(-x + x^3)^(2/3)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {4 x \left (1-x^2\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},\frac {2}{3},1,\frac {7}{6},x^2,9 x^2\right )}{\left (x^3-x\right )^{2/3}}-\frac {x \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right ) \sqrt {\frac {\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \left (x^3-x\right )^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{\sqrt [3]{x^2-1} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}}}-\frac {\sqrt {3} x^{2/3} \left (x^2-1\right )^{2/3} \arctan \left (\frac {1-\frac {4 x^{2/3}}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{2 \left (x^3-x\right )^{2/3}}-\frac {\sqrt {3} x^{2/3} \left (x^2-1\right )^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \left (x^3-x\right )^{2/3}}+\frac {x^{2/3} \left (x^2-1\right )^{2/3} \log \left (1-9 x^2\right )}{4 \left (x^3-x\right )^{2/3}}-\frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}}-\frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \log \left (2 x^{2/3}+\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}} \]