\[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {a^{\frac {1}{6}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \sqrt {-3+2 \sqrt {3}}}{\sqrt {-b \,x^{3}+a}}\right ) \sqrt {2}\, 3^{\frac {1}{4}}}{3 a^{\frac {1}{6}} b^{\frac {2}{3}}}+\frac {2 \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{3}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}+a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{3 b^{\frac {2}{3}} \sqrt {-b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right )}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x/(-b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(-b*x^3+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {\sqrt {2} a^{\frac {1}{3}} b^{\frac {4}{3}} \sqrt {\frac {\sqrt {3}}{a}} \log \left (\frac {b^{8} x^{24} + 1840 \, a b^{7} x^{21} + 67264 \, a^{2} b^{6} x^{18} + 58624 \, a^{3} b^{5} x^{15} + 504064 \, a^{4} b^{4} x^{12} - 2140160 \, a^{5} b^{3} x^{9} + 3100672 \, a^{6} b^{2} x^{6} - 1089536 \, a^{7} b x^{3} + 28672 \, a^{8} + 32 \, {\left (9 \, b^{7} x^{22} + 846 \, a b^{6} x^{19} + 4617 \, a^{2} b^{5} x^{16} - 5472 \, a^{3} b^{4} x^{13} + 43776 \, a^{4} b^{3} x^{10} - 98496 \, a^{5} b^{2} x^{7} + 59328 \, a^{6} b x^{4} - 4608 \, a^{7} x - \sqrt {3} {\left (5 \, b^{7} x^{22} + 505 \, a b^{6} x^{19} + 2130 \, a^{2} b^{5} x^{16} + 4928 \, a^{3} b^{4} x^{13} - 28688 \, a^{4} b^{3} x^{10} + 53760 \, a^{5} b^{2} x^{7} - 35200 \, a^{6} b x^{4} + 2560 \, a^{7} x\right )}\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} + 8 \, {\left (3 \, b^{7} x^{23} + 1077 \, a b^{6} x^{20} + 13320 \, a^{2} b^{5} x^{17} + 19200 \, a^{3} b^{4} x^{14} - 111360 \, a^{4} b^{3} x^{11} + 345024 \, a^{5} b^{2} x^{8} - 328704 \, a^{6} b x^{5} + 61440 \, a^{7} x^{2} - 2 \, \sqrt {3} {\left (b^{7} x^{23} + 299 \, a b^{6} x^{20} + 4260 \, a^{2} b^{5} x^{17} - 1520 \, a^{3} b^{4} x^{14} + 26720 \, a^{4} b^{3} x^{11} - 105024 \, a^{5} b^{2} x^{8} + 93184 \, a^{6} b x^{5} - 17920 \, a^{7} x^{2}\right )}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} + 2 \, \sqrt {2} {\left ({\left (b^{7} x^{22} + 1160 \, a b^{6} x^{19} + 23232 \, a^{2} b^{5} x^{16} + 53920 \, a^{3} b^{4} x^{13} - 148288 \, a^{4} b^{3} x^{10} + 586752 \, a^{5} b^{2} x^{7} - 496640 \, a^{6} b x^{4} + 38912 \, a^{7} x - \sqrt {3} {\left (b^{7} x^{22} + 632 \, a b^{6} x^{19} + 14736 \, a^{2} b^{5} x^{16} + 8416 \, a^{3} b^{4} x^{13} + 105920 \, a^{4} b^{3} x^{10} - 334848 \, a^{5} b^{2} x^{7} + 286720 \, a^{6} b x^{4} - 22528 \, a^{7} x\right )}\right )} \sqrt {-b x^{3} + a} a^{\frac {2}{3}} b^{\frac {1}{3}} + 12 \, {\left (17 \, a b^{6} x^{20} + 1014 \, a^{2} b^{5} x^{17} + 2748 \, a^{3} b^{4} x^{14} + 9632 \, a^{4} b^{3} x^{11} - 36096 \, a^{5} b^{2} x^{8} + 53376 \, a^{6} b x^{5} - 11008 \, a^{7} x^{2} - 2 \, \sqrt {3} {\left (5 \, a b^{6} x^{20} + 285 \, a^{2} b^{5} x^{17} + 1038 \, a^{3} b^{4} x^{14} - 784 \, a^{4} b^{3} x^{11} + 11424 \, a^{5} b^{2} x^{8} - 15168 \, a^{6} b x^{5} + 3200 \, a^{7} x^{2}\right )}\right )} \sqrt {-b x^{3} + a} a^{\frac {1}{3}} b^{\frac {2}{3}} + 2 \, {\left (13 \, a b^{7} x^{21} + 2090 \, a^{2} b^{6} x^{18} + 19776 \, a^{3} b^{5} x^{15} - 5216 \, a^{4} b^{4} x^{12} + 135872 \, a^{5} b^{3} x^{9} - 349824 \, a^{6} b^{2} x^{6} + 142336 \, a^{7} b x^{3} - 4096 \, a^{8} - \sqrt {3} {\left (7 \, a b^{7} x^{21} + 1250 \, a^{2} b^{6} x^{18} + 9984 \, a^{3} b^{5} x^{15} + 19456 \, a^{4} b^{4} x^{12} - 82624 \, a^{5} b^{3} x^{9} + 193920 \, a^{6} b^{2} x^{6} - 84992 \, a^{7} b x^{3} + 2048 \, a^{8}\right )}\right )} \sqrt {-b x^{3} + a}\right )} \sqrt {\frac {\sqrt {3}}{a}} - 32 \, \sqrt {3} {\left (35 \, a b^{7} x^{21} + 1141 \, a^{2} b^{6} x^{18} + 2544 \, a^{3} b^{5} x^{15} - 6760 \, a^{4} b^{4} x^{12} + 39520 \, a^{5} b^{3} x^{9} - 55680 \, a^{6} b^{2} x^{6} + 19712 \, a^{7} b x^{3} - 512 \, a^{8}\right )}}{b^{8} x^{24} - 80 \, a b^{7} x^{21} + 2368 \, a^{2} b^{6} x^{18} - 30080 \, a^{3} b^{5} x^{15} + 121984 \, a^{4} b^{4} x^{12} + 240640 \, a^{5} b^{3} x^{9} + 151552 \, a^{6} b^{2} x^{6} + 40960 \, a^{7} b x^{3} + 4096 \, a^{8}}\right ) + 4 \, \sqrt {-b} b^{\frac {2}{3}} {\left (\sqrt {3} + 3\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )}{12 \, b^{2}}, -\frac {\sqrt {2} a^{\frac {1}{3}} b^{\frac {4}{3}} \sqrt {-\frac {\sqrt {3}}{a}} \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {-b x^{3} + a} {\left (\sqrt {3} x - 3 \, x\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {-\frac {\sqrt {3}}{a}} - \sqrt {2} \sqrt {-b x^{3} + a} {\left (\sqrt {3} x^{2} + 3 \, x^{2}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} \sqrt {-\frac {\sqrt {3}}{a}} - 4 \, \sqrt {3} \sqrt {2} \sqrt {-b x^{3} + a} a \sqrt {-\frac {\sqrt {3}}{a}}}{12 \, {\left (b x^{3} - a\right )}}\right ) - 2 \, \sqrt {-b} b^{\frac {2}{3}} {\left (\sqrt {3} + 3\right )} {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )}{6 \, b^{2}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {2 \, {\left (2 \, b x^{4} - 2 \, a x - \sqrt {3} {\left (b x^{4} + 2 \, a x\right )}\right )} \sqrt {-b x^{3} + a} a^{\frac {2}{3}} + {\left (b x^{5} + 8 \, a x^{2} - \sqrt {3} {\left (b x^{5} - 4 \, a x^{2}\right )}\right )} \sqrt {-b x^{3} + a} a^{\frac {1}{3}} b^{\frac {1}{3}} + {\left (b x^{6} - 6 \, \sqrt {3} a x^{3} - 10 \, a x^{3}\right )} \sqrt {-b x^{3} + a} b^{\frac {2}{3}}}{b^{3} x^{9} - 21 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 8 \, a^{3}}, x\right ) \]