\[ \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (-x^{3}+1\right )}{6 \left (a +b \right )^{\frac {1}{3}}}-\frac {\ln \left (\left (a +b \right )^{\frac {1}{3}} x -\left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{2 \left (a +b \right )^{\frac {1}{3}}}+\frac {\arctan \left (\frac {\left (1+\frac {2 \left (a +b \right )^{\frac {1}{3}} x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3 \left (a +b \right )^{\frac {1}{3}}} \]
command
integrate(1/(-x^3+1)/(b*x^3+a)^(1/3),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \log \left (-\frac {{\left (a^{3} - 27 \, a^{2} b - 108 \, a b^{2} - 81 \, b^{3}\right )} x^{9} - 3 \, {\left (10 \, a^{3} + 54 \, a^{2} b + 45 \, a b^{2}\right )} x^{6} - 3 \, {\left (17 \, a^{3} + 18 \, a^{2} b\right )} x^{3} - a^{3} + 9 \, {\left ({\left (2 \, a^{2} + 3 \, a b\right )} x^{7} - {\left (a^{2} + 3 \, a b\right )} x^{4} - a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {1}{3}} + 9 \, {\left ({\left (a^{2} + 9 \, a b + 9 \, b^{2}\right )} x^{8} + {\left (7 \, a^{2} + 9 \, a b\right )} x^{5} + a^{2} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (3 \, {\left ({\left (4 \, a^{2} + 21 \, a b + 18 \, b^{2}\right )} x^{7} + {\left (13 \, a^{2} + 15 \, a b\right )} x^{4} + a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, {\left ({\left (a^{3} - 2 \, a^{2} b - 12 \, a b^{2} - 9 \, b^{3}\right )} x^{8} - 5 \, {\left (a^{3} + 4 \, a^{2} b + 3 \, a b^{2}\right )} x^{5} - 5 \, {\left (a^{3} + a^{2} b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left ({\left (a^{3} + 27 \, a^{2} b + 54 \, a b^{2} + 27 \, b^{3}\right )} x^{9} + 3 \, {\left (8 \, a^{3} + 18 \, a^{2} b + 9 \, a b^{2}\right )} x^{6} + 3 \, a^{3} x^{3} - a^{3}\right )} {\left (-a - b\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}}}{x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1}\right ) - 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left (-\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )} {\left (-a - b\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (a + b\right )} x + {\left (a x^{3} - a\right )} {\left (-a - b\right )}^{\frac {2}{3}}}{x^{3} - 1}\right ) + {\left (-a - b\right )}^{\frac {2}{3}} \log \left (\frac {3 \, {\left ({\left (2 \, a + 3 \, b\right )} x^{4} + a x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, {\left ({\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} x^{5} + 2 \, {\left (a^{2} + a b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left ({\left (a^{2} + 9 \, a b + 9 \, b^{2}\right )} x^{6} + {\left (7 \, a^{2} + 9 \, a b\right )} x^{3} + a^{2}\right )} {\left (-a - b\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right )}{18 \, {\left (a + b\right )}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (6 \, {\left ({\left (2 \, a^{2} + 3 \, a b\right )} x^{7} - {\left (a^{2} + 3 \, a b\right )} x^{4} - a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} - 6 \, {\left ({\left (a^{3} + 10 \, a^{2} b + 18 \, a b^{2} + 9 \, b^{3}\right )} x^{8} + {\left (7 \, a^{3} + 16 \, a^{2} b + 9 \, a b^{2}\right )} x^{5} + {\left (a^{3} + a^{2} b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left ({\left (a^{3} - 9 \, a^{2} b - 36 \, a b^{2} - 27 \, b^{3}\right )} x^{9} - 3 \, {\left (4 \, a^{3} + 18 \, a^{2} b + 15 \, a b^{2}\right )} x^{6} - 3 \, {\left (5 \, a^{3} + 6 \, a^{2} b\right )} x^{3} - a^{3}\right )} {\left (-a - b\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}}}{{\left (a^{3} + 27 \, a^{2} b + 54 \, a b^{2} + 27 \, b^{3}\right )} x^{9} + 3 \, {\left (8 \, a^{3} + 18 \, a^{2} b + 9 \, a b^{2}\right )} x^{6} + 3 \, a^{3} x^{3} - a^{3}}\right ) - 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left (-\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )} {\left (-a - b\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (a + b\right )} x + {\left (a x^{3} - a\right )} {\left (-a - b\right )}^{\frac {2}{3}}}{x^{3} - 1}\right ) + {\left (-a - b\right )}^{\frac {2}{3}} \log \left (\frac {3 \, {\left ({\left (2 \, a + 3 \, b\right )} x^{4} + a x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, {\left ({\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} x^{5} + 2 \, {\left (a^{2} + a b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left ({\left (a^{2} + 9 \, a b + 9 \, b^{2}\right )} x^{6} + {\left (7 \, a^{2} + 9 \, a b\right )} x^{3} + a^{2}\right )} {\left (-a - b\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right )}{18 \, {\left (a + b\right )}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]