\[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \left (b x +a \right )^{\frac {5}{2}}}{9 a \,x^{\frac {9}{2}}}+\frac {2 \left (4 A b -9 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{63 a^{2} x^{\frac {7}{2}}}-\frac {4 b \left (4 A b -9 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{315 a^{3} x^{\frac {5}{2}}} \]
command
integrate((b*x+a)**(3/2)*(B*x+A)/x**(11/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ A \left (- \frac {70 a^{6} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {240 a^{5} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {276 a^{4} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {104 a^{3} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {6 a^{2} b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {24 a b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {16 b^{\frac {21}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}}\right ) + B \left (- \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{7 x^{3}} - \frac {16 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{35 x^{2}} - \frac {2 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a x} + \frac {4 b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{2}}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________