\[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx \]
Optimal antiderivative \[ -\frac {3 b^{3} \left (b x +a \right )^{\frac {3}{2}}}{64 x^{4}}-\frac {3 b^{2} \left (b x +a \right )^{\frac {5}{2}}}{40 x^{5}}-\frac {3 b \left (b x +a \right )^{\frac {7}{2}}}{28 x^{6}}-\frac {\left (b x +a \right )^{\frac {9}{2}}}{7 x^{7}}-\frac {9 b^{7} \arctanh \! \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {5}{2}}}-\frac {3 b^{4} \sqrt {b x +a}}{128 x^{3}}-\frac {3 b^{5} \sqrt {b x +a}}{512 a \,x^{2}}+\frac {9 b^{6} \sqrt {b x +a}}{1024 a^{2} x} \]
command
integrate((b*x+a)**(9/2)/x**8,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {a^{5}}{7 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {23 a^{4} \sqrt {b}}{28 x^{\frac {13}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {541 a^{3} b^{\frac {3}{2}}}{280 x^{\frac {11}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5249 a^{2} b^{\frac {5}{2}}}{2240 x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {6653 a b^{\frac {7}{2}}}{4480 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {1027 b^{\frac {9}{2}}}{2560 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {3 b^{\frac {11}{2}}}{1024 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {9 b^{\frac {13}{2}}}{1024 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {9 b^{7} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{1024 a^{\frac {5}{2}}} \]