\[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \left (b x +a \right )^{\frac {7}{2}}}{9 a \,x^{\frac {9}{2}}}+\frac {2 \left (2 A b -9 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{63 a^{2} x^{\frac {7}{2}}} \]
command
integrate((b*x+a)**(5/2)*(B*x+A)/x**(11/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ A \left (- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{9 x^{4}} - \frac {38 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{63 x^{3}} - \frac {10 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{21 x^{2}} - \frac {2 b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{63 a x} + \frac {4 b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{63 a^{2}}\right ) + B \left (- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{7 x^{3}} - \frac {6 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{7 x^{2}} - \frac {6 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{7 x} - \frac {2 b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{7 a}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________