\[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (7 A b -B a \right ) \arctan \left (\frac {b^{\frac {1}{6}} \sqrt {x}}{a^{\frac {1}{6}}}\right )}{9 a^{\frac {13}{6}} b^{\frac {5}{6}}}-\frac {\left (7 A b -B a \right ) \arctan \left (-\sqrt {3}+\frac {2 b^{\frac {1}{6}} \sqrt {x}}{a^{\frac {1}{6}}}\right )}{18 a^{\frac {13}{6}} b^{\frac {5}{6}}}-\frac {\left (7 A b -B a \right ) \arctan \left (\sqrt {3}+\frac {2 b^{\frac {1}{6}} \sqrt {x}}{a^{\frac {1}{6}}}\right )}{18 a^{\frac {13}{6}} b^{\frac {5}{6}}}-\frac {\left (7 A b -B a \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x -a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {3}\, \sqrt {x}\right ) \sqrt {3}}{36 a^{\frac {13}{6}} b^{\frac {5}{6}}}+\frac {\left (7 A b -B a \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x +a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {3}\, \sqrt {x}\right ) \sqrt {3}}{36 a^{\frac {13}{6}} b^{\frac {5}{6}}}+\frac {-7 A b +B a}{3 a^{2} b \sqrt {x}}+\frac {A b -B a}{3 a b \left (b \,x^{3}+a \right ) \sqrt {x}} \]
command
integrate((B*x**3+A)/x**(3/2)/(b*x**3+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________