\[ \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {21 b \,x^{\frac {3}{2}}}{4 a^{4}}+\frac {63 x^{\frac {5}{2}}}{20 a^{3}}-\frac {x^{\frac {9}{2}}}{2 a \left (a x +b \right )^{2}}-\frac {9 x^{\frac {7}{2}}}{4 a^{2} \left (a x +b \right )}-\frac {63 b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b}}\right )}{4 a^{\frac {11}{2}}}+\frac {63 b^{2} \sqrt {x}}{4 a^{5}} \]
command
integrate(x**(3/2)/(a+b/x)**3,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 {5}{2}}}{5 a^{3}} & \text {for}\: b = 0 \\\frac {2 x^{\frac {11}{2}}}{11 b^{3}} & \text {for}\: a = 0 \\\frac {16 a^{5} x^{\frac {9}{2}} \sqrt {- \frac {b}{a}}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} - \frac {48 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {b}{a}}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} + \frac {336 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} + \frac {1050 a^{2} b^{3} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} - \frac {315 a^{2} b^{3} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} + \frac {315 a^{2} b^{3} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} + \frac {630 a b^{4} \sqrt {x} \sqrt {- \frac {b}{a}}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} - \frac {630 a b^{4} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} + \frac {630 a b^{4} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} - \frac {315 b^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} + \frac {315 b^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{40 a^{8} x^{2} \sqrt {- \frac {b}{a}} + 80 a^{7} b x \sqrt {- \frac {b}{a}} + 40 a^{6} b^{2} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________