\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {5}{3 b^{2} x^{\frac {3}{2}}}+\frac {1}{b \,x^{\frac {3}{2}} \left (a x +b \right )}+\frac {5 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b}}\right )}{b^{\frac {7}{2}}}+\frac {5 a}{b^{3} \sqrt {x}} \]
command
integrate(1/(a+b/x)**2/x**(9/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2}{7 a^{2} x^{\frac {7}{2}}} & \text {for}\: b = 0 \\\frac {15 a^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {15 a^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {30 a^{2} x^{2} \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {20 a b x \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {4 b^{2} \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________