\[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{7 a \,x^{\frac {7}{2}} \sqrt {b x +a}}-\frac {2 \left (8 A b -7 B a \right )}{7 a^{2} x^{\frac {5}{2}} \sqrt {b x +a}}+\frac {12 \left (8 A b -7 B a \right ) \sqrt {b x +a}}{35 a^{3} x^{\frac {5}{2}}}-\frac {16 b \left (8 A b -7 B a \right ) \sqrt {b x +a}}{35 a^{4} x^{\frac {3}{2}}}+\frac {32 b^{2} \left (8 A b -7 B a \right ) \sqrt {b x +a}}{35 a^{5} \sqrt {x}} \]
command
integrate((B*x+A)/x**(9/2)/(b*x+a)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ A \left (- \frac {10 a^{7} b^{\frac {33}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} - \frac {14 a^{6} b^{\frac {35}{2}} x \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} - \frac {14 a^{5} b^{\frac {37}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {70 a^{4} b^{\frac {39}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {560 a^{3} b^{\frac {41}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {1120 a^{2} b^{\frac {43}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {896 a b^{\frac {45}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {256 b^{\frac {47}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}}\right ) + B \left (- \frac {2 a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {10 a^{3} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {60 a^{2} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {80 a b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {32 b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________