\[ \int \frac {\sqrt [3]{a+b x^3}}{x^2 \left (a d-b d x^3\right )} \, dx \]
Optimal antiderivative \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{a d x}+\frac {b^{\frac {1}{3}} \ln \left (-b d \,x^{3}+a d \right ) 2^{\frac {1}{3}}}{6 a d}-\frac {b^{\frac {1}{3}} \ln \left (2^{\frac {1}{3}} b^{\frac {1}{3}} x -\left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{2 a d}-\frac {2^{\frac {1}{3}} b^{\frac {1}{3}} \arctan \left (\frac {\left (1+\frac {2 \,2^{\frac {1}{3}} b^{\frac {1}{3}} x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3 a d} \]
command
integrate((b*x^3+a)^(1/3)/x^2/(-b*d*x^3+a*d),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (19 \, b^{2} x^{8} + 16 \, a b x^{5} + a^{2} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} + 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (5 \, b^{2} x^{7} - 4 \, a b x^{4} - a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} + \sqrt {3} {\left (71 \, b^{3} x^{9} + 111 \, a b^{2} x^{6} + 33 \, a^{2} b x^{3} + a^{3}\right )}}{3 \, {\left (109 \, b^{3} x^{9} + 105 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} - a^{3}\right )}}\right ) - 2 \cdot 2^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} b x^{2} + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x + 2^{\frac {2}{3}} {\left (b x^{3} - a\right )} \left (-b\right )^{\frac {2}{3}}}{b x^{3} - a}\right ) + 2^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, b x^{4} + a x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (19 \, b^{2} x^{6} + 16 \, a b x^{3} + a^{2}\right )} \left (-b\right )^{\frac {1}{3}} + 12 \, {\left (2 \, b^{2} x^{5} + a b x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{2} x^{6} - 2 \, a b x^{3} + a^{2}}\right ) + 18 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{18 \, a d x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]