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