\[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx \]
Optimal antiderivative \[ \frac {\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}{a}-\frac {\left (3 a^{2} b +b^{2}\right ) \arctan \left (\frac {\sqrt {3}\, a x}{a x +2 \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3 a^{3}}-\frac {\sqrt {-3-3 i \sqrt {3}}\, b \left (a^{2}+b \right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x}{a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x -2 \left (-1\right )^{\frac {1}{3}} \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\right ) \sqrt {2}}{2 a^{\frac {5}{3}}}+\frac {\left (-3 a^{2} b -b^{2}\right ) \ln \left (-a x +\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}\right )}{3 a^{3}}+\frac {i \left (i b \left (a^{2}+b \right )^{\frac {1}{3}}+\sqrt {3}\, b \left (a^{2}+b \right )^{\frac {1}{3}}\right ) \ln \left (a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x +\left (-1\right )^{\frac {1}{3}} \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}\right )}{2 a^{\frac {5}{3}}}+\frac {\left (3 a^{2} b +b^{2}\right ) \ln \left (a^{2} x^{2}+a x \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}+\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {2}{3}}\right )}{6 a^{3}}+\frac {\left (b \left (a^{2}+b \right )^{\frac {1}{3}}-i \sqrt {3}\, b \left (a^{2}+b \right )^{\frac {1}{3}}\right ) \ln \left (a^{\frac {2}{3}} \left (a^{2}+b \right )^{\frac {2}{3}} x^{2}-\left (-1\right )^{\frac {1}{3}} a^{\frac {1}{3}} \left (a^{2}+b \right )^{\frac {1}{3}} x \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}+\left (-1\right )^{\frac {2}{3}} \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {2}{3}}\right )}{4 a^{\frac {5}{3}}} \]
command
integrate((a^3*x^3+b^2*x^2)^(1/3)/(a*x-b),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{2} b + b^{2}\right )} \log \left ({\left | -{\left (a^{3} + a b\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{4} + a^{2} b} - \frac {\sqrt {3} {\left (a^{3} + a b\right )}^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} + a b\right )}^{\frac {1}{3}}}\right )}{a^{2}} + \frac {{\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} x}{a} - \frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} b \log \left ({\left (a^{3} + a b\right )}^{\frac {2}{3}} + {\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\sqrt {3} {\left (3 \, a^{2} b + b^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{3 \, a^{3}} + \frac {{\left (3 \, a^{2} b + b^{2}\right )} \log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{6 \, a^{3}} - \frac {{\left (3 \, a^{2} b + b^{2}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________