\[ \int \frac {x^{7/2} \left (A+B x^3\right )}{a+b x^3} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A b -B a \right ) x^{\frac {3}{2}}}{3 b^{2}}+\frac {2 B \,x^{\frac {9}{2}}}{9 b}-\frac {2 \left (A b -B a \right ) \arctan \left (\frac {x^{\frac {3}{2}} \sqrt {b}}{\sqrt {a}}\right ) \sqrt {a}}{3 b^{\frac {5}{2}}} \]
command
integrate(x**(7/2)*(B*x**3+A)/(b*x**3+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {9}{2}}}{9}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {9}{2}}}{9} + \frac {2 B x^{\frac {15}{2}}}{15}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {9}{2}}}{9}}{b} & \text {for}\: a = 0 \\- \frac {A a \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 b^{2} \sqrt {- \frac {a}{b}}} + \frac {A a \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 b^{2} \sqrt {- \frac {a}{b}}} + \frac {A a \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{3 b^{2} \sqrt {- \frac {a}{b}}} - \frac {A a \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{3 b^{2} \sqrt {- \frac {a}{b}}} + \frac {2 A x^{\frac {3}{2}}}{3 b} + \frac {B a^{2} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 b^{3} \sqrt {- \frac {a}{b}}} - \frac {B a^{2} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 b^{3} \sqrt {- \frac {a}{b}}} - \frac {B a^{2} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{3 b^{3} \sqrt {- \frac {a}{b}}} + \frac {B a^{2} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{3 b^{3} \sqrt {- \frac {a}{b}}} - \frac {2 B a x^{\frac {3}{2}}}{3 b^{2}} + \frac {2 B x^{\frac {9}{2}}}{9 b} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________