\[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx \]
Optimal antiderivative \[ -\frac {\left (1-x \right ) \EllipticF \left (\frac {1-x -\sqrt {3}}{1-x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (e +f \left (1-\sqrt {3}\right )\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}+x +1}{\left (1-x +\sqrt {3}\right )^{2}}}\, 3^{\frac {1}{4}}}{3 \sqrt {-x^{3}+1}\, \sqrt {\frac {1-x}{\left (1-x +\sqrt {3}\right )^{2}}}}-\frac {\arctan \left (\frac {\left (1-x \right ) \sqrt {3+2 \sqrt {3}}}{\sqrt {-x^{3}+1}}\right ) \left (e +f +f \sqrt {3}\right )}{\sqrt {9+6 \sqrt {3}}} \]
command
integrate((f*x+e)/(1-x+3^(1/2))/(-x^3+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {1}{12} \, \sqrt {-6 \, f e - 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} + 3 \, e^{2}} \log \left (-\frac {2 \, f^{2} x^{8} + 32 \, f^{2} x^{7} + 224 \, f^{2} x^{6} + 32 \, f^{2} x^{5} + 224 \, f^{2} x^{4} - 448 \, f^{2} x^{3} + 128 \, f^{2} x^{2} - 256 \, f^{2} x + 4 \, {\left (f x^{6} + 18 \, f x^{5} + 12 \, f x^{4} + 40 \, f x^{3} - 36 \, f x^{2} + 24 \, f x - 2 \, {\left (x^{6} + 9 \, x^{5} + 21 \, x^{4} + 4 \, x^{3} - 12 \, x + 4\right )} e + \sqrt {3} {\left (f x^{6} + 6 \, f x^{5} + 24 \, f x^{4} - 8 \, f x^{3} + 12 \, f x^{2} - 24 \, f x - {\left (x^{6} + 12 \, x^{5} + 18 \, x^{4} + 16 \, x^{3} - 12 \, x^{2} - 8\right )} e + 16 \, f\right )} - 32 \, f\right )} \sqrt {-x^{3} + 1} \sqrt {-6 \, f e - 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} + 3 \, e^{2}} + 224 \, f^{2} - {\left (x^{8} + 16 \, x^{7} + 112 \, x^{6} + 16 \, x^{5} + 112 \, x^{4} - 224 \, x^{3} + 64 \, x^{2} - 128 \, x + 112\right )} e^{2} - 2 \, {\left (f x^{8} + 16 \, f x^{7} + 112 \, f x^{6} + 16 \, f x^{5} + 112 \, f x^{4} - 224 \, f x^{3} + 64 \, f x^{2} - 128 \, f x + 112 \, f\right )} e + 16 \, \sqrt {3} {\left (2 \, f^{2} x^{7} + 4 \, f^{2} x^{6} + 12 \, f^{2} x^{5} - 10 \, f^{2} x^{4} + 4 \, f^{2} x^{3} - 12 \, f^{2} x^{2} + 8 \, f^{2} x - 8 \, f^{2} - {\left (x^{7} + 2 \, x^{6} + 6 \, x^{5} - 5 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} + 4 \, x - 4\right )} e^{2} - 2 \, {\left (f x^{7} + 2 \, f x^{6} + 6 \, f x^{5} - 5 \, f x^{4} + 2 \, f x^{3} - 6 \, f x^{2} + 4 \, f x - 4 \, f\right )} e\right )}}{x^{8} - 8 \, x^{7} + 16 \, x^{6} + 16 \, x^{5} - 56 \, x^{4} - 32 \, x^{3} + 64 \, x^{2} + 64 \, x + 16}\right ), -\frac {1}{6} \, \sqrt {6 \, f e + 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} - 3 \, e^{2}} \arctan \left (\frac {{\left (3 \, f x^{2} + 6 \, f x - 6 \, {\left (x - 1\right )} e - \sqrt {3} {\left (f x^{2} - 2 \, f x + {\left (x^{2} + 4 \, x - 2\right )} e + 4 \, f\right )}\right )} \sqrt {-x^{3} + 1} \sqrt {6 \, f e + 2 \, \sqrt {3} {\left (f^{2} - f e + e^{2}\right )} - 3 \, e^{2}}}{6 \, {\left (2 \, f^{2} x^{3} - 2 \, f^{2} - {\left (x^{3} - 1\right )} e^{2} - 2 \, {\left (f x^{3} - f\right )} e\right )}}\right )\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (f x^{2} + {\left (e - f\right )} x + \sqrt {3} {\left (f x + e\right )} - e\right )} \sqrt {-x^{3} + 1}}{x^{5} - 2 \, x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 2}, x\right ) \]