\[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx \]
Optimal antiderivative \[ \frac {\left (a d +4 b c \right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}} x^{2}}{8 c}+\frac {a \left (c +\frac {d}{x^{2}}\right )^{\frac {5}{2}} x^{4}}{4 c}+\frac {3 d \left (a d +4 b c \right ) \arctanh \left (\frac {\sqrt {c +\frac {d}{x^{2}}}}{\sqrt {c}}\right )}{8 \sqrt {c}}-\frac {3 d \left (a d +4 b c \right ) \sqrt {c +\frac {d}{x^{2}}}}{8 c} \]
command
integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a c^{2} x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 a c \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{\frac {3}{2}} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} + \frac {a d^{\frac {3}{2}} x}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 \sqrt {c}} + \frac {3 b \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2} + \frac {b c \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} - \frac {b c \sqrt {d} x}{\sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b d^{\frac {3}{2}}}{x \sqrt {\frac {c x^{2}}{d} + 1}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________