\[ \int \frac {b+a x}{(-b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, a x}{a x +2 \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{a}+\frac {\sqrt {-6+6 i \sqrt {3}}\, \arctan \left (\frac {\sqrt {3}\, a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x}{a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x -2 \left (-1\right )^{\frac {1}{3}} \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}}}-\frac {\ln \left (-a x +\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}\right )}{a}-\frac {i \left (-i+\sqrt {3}\right ) \ln \left (a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x +\left (-1\right )^{\frac {1}{3}} \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}\right )}{a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}}}+\frac {\ln \left (a^{2} x^{2}+a x \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}+\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {2}{3}}\right )}{2 a}+\frac {\left (1+i \sqrt {3}\right ) \ln \left (a^{\frac {2}{3}} \left (a^{2}+b \right )^{\frac {2}{3}} x^{2}-\left (-1\right )^{\frac {1}{3}} a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}+\left (-1\right )^{\frac {2}{3}} \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}}} \]
command
integrate((a*x+b)/(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
\[ \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} + a b\right )}^{\frac {1}{3}}}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} - \frac {\log \left ({\left (a^{3} + a b\right )}^{\frac {2}{3}} + {\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | -{\left (a^{3} + a b\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} - \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a} + \frac {\log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a} - \frac {\log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________