\[ \int \frac {x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 \,2^{\frac {2}{3}} \arctanh \left (\frac {\left (1-2^{\frac {1}{3}} x \right ) \sqrt {3}}{\sqrt {x^{3}-1}}\right ) \sqrt {3}}{9}+\frac {2 \left (1-x \right ) \EllipticF \left (\frac {1-x +\sqrt {3}}{1-x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}+x +1}{\left (1-x -\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {x^{3}-1}\, \sqrt {\frac {-1+x}{\left (1-x -\sqrt {3}\right )^{2}}}} \]
command
integrate(x/(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
\[ \frac {1}{18} \, \sqrt {3} 2^{\frac {2}{3}} \log \left (\frac {x^{18} + 1440 \, x^{15} + 17400 \, x^{12} - 21056 \, x^{9} - 10368 \, x^{6} + 15360 \, x^{3} + 2 \, \sqrt {3} 2^{\frac {2}{3}} {\left (126 \, x^{14} + 2664 \, x^{11} - 4608 \, x^{5} + 2304 \, x^{2} + 2^{\frac {2}{3}} {\left (x^{16} + 310 \, x^{13} + 2332 \, x^{10} - 2656 \, x^{7} - 256 \, x^{4} + 512 \, x\right )} + 2^{\frac {1}{3}} {\left (17 \, x^{15} + 1058 \, x^{12} + 2528 \, x^{9} - 5408 \, x^{6} + 2560 \, x^{3} - 512\right )}\right )} \sqrt {x^{3} - 1} + 24 \cdot 2^{\frac {2}{3}} {\left (x^{17} + 121 \, x^{14} + 478 \, x^{11} - 1144 \, x^{8} + 608 \, x^{5} - 64 \, x^{2}\right )} + 48 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} + 176 \, x^{13} + 83 \, x^{10} - 680 \, x^{7} + 544 \, x^{4} - 128 \, x\right )} - 2048}{x^{18} - 24 \, x^{15} + 240 \, x^{12} - 1280 \, x^{9} + 3840 \, x^{6} - 6144 \, x^{3} + 4096}\right ) - \frac {2}{3} \, {\rm weierstrassPInverse}\left (0, 4, x\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (x^{3} + 2^{\frac {2}{3}} x^{2} + 2 \cdot 2^{\frac {1}{3}} x\right )} \sqrt {x^{3} - 1}}{x^{6} - 5 \, x^{3} + 4}, x\right ) \]