\[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx \]
Optimal antiderivative \[ \frac {3 b^{2} x^{\frac {3}{2}}}{a^{4}}-\frac {9 b \,x^{\frac {5}{2}}}{5 a^{3}}+\frac {9 x^{\frac {7}{2}}}{7 a^{2}}-\frac {x^{\frac {9}{2}}}{a \left (a x +b \right )}+\frac {9 b^{\frac {7}{2}} \arctan \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b}}\right )}{a^{\frac {11}{2}}}-\frac {9 b^{3} \sqrt {x}}{a^{5}} \]
command
integrate(x**(5/2)/(a+b/x)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } x^{\frac {11}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {11}{2}}}{11 b^{2}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a^{2}} & \text {for}\: b = 0 \\\frac {20 a^{5} x^{\frac {9}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {36 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {84 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {420 a^{2} b^{3} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {630 a b^{4} \sqrt {x} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {315 a b^{4} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {315 a b^{4} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {315 b^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {315 b^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________