\[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \left (b x +a \right )^{\frac {3}{2}}}{9 a \,x^{\frac {9}{2}}}+\frac {2 \left (2 A b -3 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{21 a^{2} x^{\frac {7}{2}}}-\frac {8 b \left (2 A b -3 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{105 a^{3} x^{\frac {5}{2}}}+\frac {16 b^{2} \left (2 A b -3 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{315 a^{4} x^{\frac {3}{2}}} \]
command
integrate((B*x+A)*(b*x+a)**(1/2)/x**(11/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ A \left (- \frac {70 a^{7} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} - \frac {220 a^{6} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} - \frac {228 a^{5} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} - \frac {80 a^{4} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {10 a^{3} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {60 a^{2} b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {80 a b^{\frac {31}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {32 b^{\frac {33}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}}\right ) + B \left (- \frac {30 a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {66 a^{4} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 a^{3} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {6 a^{2} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 a b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {16 b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________