\[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {-3+3 i \sqrt {3}}\, \arctan \left (\frac {\sqrt {3}\, a^{\frac {1}{3}} \left (a^{2}-b \right )^{\frac {1}{3}} x}{a^{\frac {1}{3}} \left (a^{2}-b \right )^{\frac {1}{3}} x -2 \left (-1\right )^{\frac {1}{3}} \left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {1}{3}}}\right ) \sqrt {2}}{2 a^{\frac {1}{3}} \left (a^{2}-b \right )^{\frac {1}{3}} b}-\frac {i \left (-i+\sqrt {3}\right ) \ln \left (a^{\frac {1}{3}} \left (a^{2}-b \right )^{\frac {1}{3}} x +\left (-1\right )^{\frac {1}{3}} \left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {1}{3}}\right )}{2 a^{\frac {1}{3}} \left (a^{2}-b \right )^{\frac {1}{3}} b}+\frac {\left (1+i \sqrt {3}\right ) \ln \left (a^{\frac {2}{3}} \left (a^{2}-b \right )^{\frac {2}{3}} x^{2}-\left (-1\right )^{\frac {1}{3}} a^{\frac {1}{3}} \left (a^{2}-b \right )^{\frac {1}{3}} x \left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {1}{3}}+\left (-1\right )^{\frac {2}{3}} \left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {2}{3}}\right )}{4 a^{\frac {1}{3}} \left (a^{2}-b \right )^{\frac {1}{3}} b} \]
command
integrate(1/(a*x-b)/(a^3*x^3-b^2*x^2)^(1/3),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - a b\right )}^{\frac {1}{3}}}\right )}{\sqrt {3} a^{3} b - \sqrt {3} a b^{2}} - \frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left ({\left (a^{3} - a b\right )}^{\frac {2}{3}} + {\left (a^{3} - a b\right )}^{\frac {1}{3}} {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, {\left (a^{3} b - a b^{2}\right )}} + \frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left ({\left | -{\left (a^{3} - a b\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{3} b - a b^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________