\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b}}\right )}{4 a^{\frac {3}{2}} b^{\frac {3}{2}}}-\frac {\sqrt {x}}{2 a \left (a x +b \right )^{2}}+\frac {\sqrt {x}}{4 a b \left (a x +b \right )} \]
command
integrate(1/(a+b/x)**3/x**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b^{3}} & \text {for}\: a = 0 \\- \frac {2}{3 a^{3} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\\frac {2 a^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} + \frac {a^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} - \frac {a^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} - \frac {2 a b \sqrt {x} \sqrt {- \frac {b}{a}}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} + \frac {2 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} - \frac {2 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} + \frac {b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} - \frac {b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{8 a^{4} b x^{2} \sqrt {- \frac {b}{a}} + 16 a^{3} b^{2} x \sqrt {- \frac {b}{a}} + 8 a^{2} b^{3} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________