\[ \int \frac {\sqrt [3]{b x+a x^3} \left (b+a x^4\right )}{x^4} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{3} a +b x \right )^{\frac {1}{3}} \left (4 a \,x^{4}-3 a \,x^{2}-3 b \right )}{8 x^{3}}-\frac {a^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3}\, a^{\frac {1}{3}} x}{a^{\frac {1}{3}} x +2 \left (x^{3} a +b x \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{6}-\frac {a^{\frac {1}{3}} b \ln \left (-a^{\frac {1}{3}} x +\left (x^{3} a +b x \right )^{\frac {1}{3}}\right )}{6}+\frac {a^{\frac {1}{3}} b \ln \left (a^{\frac {2}{3}} x^{2}+a^{\frac {1}{3}} x \left (x^{3} a +b x \right )^{\frac {1}{3}}+\left (x^{3} a +b x \right )^{\frac {2}{3}}\right )}{12} \]
command
integrate((a*x^3+b*x)^(1/3)*(a*x^4+b)/x^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {12 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} a b x^{2} + 4 \, \sqrt {3} a^{\frac {1}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2 \, a^{\frac {1}{3}} b^{2} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {2}{3}} + {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} b^{2} \log \left ({\left | {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) - 9 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {4}{3}} b}{24 \, b} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________