\[ \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{5 a \,x^{\frac {5}{2}}}-\frac {2 \left (A b -B a \right ) \arctan \left (\frac {b^{\frac {1}{6}} \sqrt {x}}{a^{\frac {1}{6}}}\right )}{3 a^{\frac {11}{6}} b^{\frac {1}{6}}}-\frac {\left (A b -B a \right ) \arctan \left (-\sqrt {3}+\frac {2 b^{\frac {1}{6}} \sqrt {x}}{a^{\frac {1}{6}}}\right )}{3 a^{\frac {11}{6}} b^{\frac {1}{6}}}-\frac {\left (A b -B a \right ) \arctan \left (\sqrt {3}+\frac {2 b^{\frac {1}{6}} \sqrt {x}}{a^{\frac {1}{6}}}\right )}{3 a^{\frac {11}{6}} b^{\frac {1}{6}}}+\frac {\left (A b -B a \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x -a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {3}\, \sqrt {x}\right ) \sqrt {3}}{6 a^{\frac {11}{6}} b^{\frac {1}{6}}}-\frac {\left (A b -B a \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x +a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {3}\, \sqrt {x}\right ) \sqrt {3}}{6 a^{\frac {11}{6}} b^{\frac {1}{6}}} \]
command
integrate((B*x**3+A)/x**(7/2)/(b*x**3+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} + 2 B \sqrt {x}}{a} & \text {for}\: b = 0 \\- \frac {2 A}{5 a x^{\frac {5}{2}}} + \frac {A b \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 a^{2}} - \frac {A b \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 a^{2}} + \frac {A b \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2}} - \frac {A b \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2}} - \frac {\sqrt {3} A b \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{3 a^{2}} - \frac {\sqrt {3} A b \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a^{2}} - \frac {B \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 a} + \frac {B \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 a} - \frac {B \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a} + \frac {B \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a} + \frac {\sqrt {3} B \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{3 a} + \frac {\sqrt {3} B \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________