\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e +2^{\frac {2}{3}} f \right ) \arctan \left (\frac {\left (1-2^{\frac {1}{3}} x \right ) \sqrt {3}}{\sqrt {-x^{3}+1}}\right ) \sqrt {3}}{9}-\frac {2 \left (2^{\frac {1}{3}} e -f \right ) \left (1-x \right ) \EllipticF \left (\frac {1-x -\sqrt {3}}{1-x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}+x +1}{\left (1-x +\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {-x^{3}+1}\, \sqrt {\frac {1-x}{\left (1-x +\sqrt {3}\right )^{2}}}} \]
command
integrate((f*x+e)/(2^(2/3)-x)/(-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}{18} \, \sqrt {3} \sqrt {-2 \cdot 2^{\frac {2}{3}} f e - 2 \cdot 2^{\frac {1}{3}} f^{2} - e^{2}} \log \left (\frac {4 \, f^{3} x^{18} + 5760 \, f^{3} x^{15} + 69600 \, f^{3} x^{12} - 84224 \, f^{3} x^{9} - 41472 \, f^{3} x^{6} + 61440 \, f^{3} x^{3} - 8192 \, f^{3} + 4 \, \sqrt {3} {\left (252 \, f^{2} x^{14} + 5328 \, f^{2} x^{11} - 9216 \, f^{2} x^{5} + 4608 \, f^{2} x^{2} + {\left (17 \, x^{15} + 1058 \, x^{12} + 2528 \, x^{9} - 5408 \, x^{6} + 2560 \, x^{3} - 512\right )} e^{2} - 2 \, {\left (f x^{16} + 310 \, f x^{13} + 2332 \, f x^{10} - 2656 \, f x^{7} - 256 \, f x^{4} + 512 \, f x\right )} e + 2^{\frac {2}{3}} {\left (2 \, f^{2} x^{16} + 620 \, f^{2} x^{13} + 4664 \, f^{2} x^{10} - 5312 \, f^{2} x^{7} - 512 \, f^{2} x^{4} + 1024 \, f^{2} x + 9 \, {\left (7 \, x^{14} + 148 \, x^{11} - 256 \, x^{5} + 128 \, x^{2}\right )} e^{2} - {\left (17 \, f x^{15} + 1058 \, f x^{12} + 2528 \, f x^{9} - 5408 \, f x^{6} + 2560 \, f x^{3} - 512 \, f\right )} e\right )} + 2^{\frac {1}{3}} {\left (34 \, f^{2} x^{15} + 2116 \, f^{2} x^{12} + 5056 \, f^{2} x^{9} - 10816 \, f^{2} x^{6} + 5120 \, f^{2} x^{3} - 1024 \, f^{2} + {\left (x^{16} + 310 \, x^{13} + 2332 \, x^{10} - 2656 \, x^{7} - 256 \, x^{4} + 512 \, x\right )} e^{2} - 18 \, {\left (7 \, f x^{14} + 148 \, f x^{11} - 256 \, f x^{5} + 128 \, f x^{2}\right )} e\right )}\right )} \sqrt {-x^{3} + 1} \sqrt {-2 \cdot 2^{\frac {2}{3}} f e - 2 \cdot 2^{\frac {1}{3}} f^{2} - e^{2}} + {\left (x^{18} + 1440 \, x^{15} + 17400 \, x^{12} - 21056 \, x^{9} - 10368 \, x^{6} + 15360 \, x^{3} - 2048\right )} e^{3} + 24 \cdot 2^{\frac {2}{3}} {\left (4 \, f^{3} x^{17} + 484 \, f^{3} x^{14} + 1912 \, f^{3} x^{11} - 4576 \, f^{3} x^{8} + 2432 \, f^{3} x^{5} - 256 \, f^{3} x^{2} + {\left (x^{17} + 121 \, x^{14} + 478 \, x^{11} - 1144 \, x^{8} + 608 \, x^{5} - 64 \, x^{2}\right )} e^{3}\right )} + 48 \cdot 2^{\frac {1}{3}} {\left (20 \, f^{3} x^{16} + 704 \, f^{3} x^{13} + 332 \, f^{3} x^{10} - 2720 \, f^{3} x^{7} + 2176 \, f^{3} x^{4} - 512 \, f^{3} x + {\left (5 \, x^{16} + 176 \, x^{13} + 83 \, x^{10} - 680 \, x^{7} + 544 \, x^{4} - 128 \, x\right )} e^{3}\right )}}{x^{18} - 24 \, x^{15} + 240 \, x^{12} - 1280 \, x^{9} + 3840 \, x^{6} - 6144 \, x^{3} + 4096}\right ), -\frac {1}{9} \, \sqrt {3} \sqrt {2 \cdot 2^{\frac {2}{3}} f e + 2 \cdot 2^{\frac {1}{3}} f^{2} + e^{2}} \arctan \left (\frac {\sqrt {3} {\left (4 \, f^{2} x^{5} - 4 \, f^{2} x^{2} - {\left (5 \, x^{3} - 2\right )} e^{2} - 2 \, {\left (7 \, f x^{4} - 4 \, f x\right )} e + 2^{\frac {2}{3}} {\left (14 \, f^{2} x^{4} - 8 \, f^{2} x + {\left (x^{5} - x^{2}\right )} e^{2} + {\left (5 \, f x^{3} - 2 \, f\right )} e\right )} - 2^{\frac {1}{3}} {\left (10 \, f^{2} x^{3} - 4 \, f^{2} - {\left (7 \, x^{4} - 4 \, x\right )} e^{2} + 2 \, {\left (f x^{5} - f x^{2}\right )} e\right )}\right )} \sqrt {-x^{3} + 1} \sqrt {2 \cdot 2^{\frac {2}{3}} f e + 2 \cdot 2^{\frac {1}{3}} f^{2} + e^{2}}}{6 \, {\left (8 \, f^{3} x^{6} - 12 \, f^{3} x^{3} + 4 \, f^{3} + {\left (2 \, x^{6} - 3 \, x^{3} + 1\right )} e^{3}\right )}}\right )\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (f x^{3} + e x^{2} + 2^{\frac {2}{3}} {\left (f x^{2} + e x\right )} + 2 \cdot 2^{\frac {1}{3}} {\left (f x + e\right )}\right )} \sqrt {-x^{3} + 1}}{x^{6} - 5 \, x^{3} + 4}, x\right ) \]