\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \, dx \]
Optimal antiderivative \[ \frac {2 d^{\frac {5}{2}} \arctan \! \left (\frac {\sqrt {d}\, \sqrt {a +\frac {b}{x}}}{\sqrt {-a d +b c}}\right )}{c^{2} \left (-a d +b c \right )^{\frac {3}{2}}}-\frac {\left (2 a d +3 b c \right ) \arctanh \! \left (\frac {\sqrt {a +\frac {b}{x}}}{\sqrt {a}}\right )}{a^{\frac {5}{2}} c^{2}}+\frac {b \left (-a d +3 b c \right )}{a^{2} c \left (-a d +b c \right ) \sqrt {a +\frac {b}{x}}}+\frac {x}{a c \sqrt {a +\frac {b}{x}}} \]
command
integrate(1/(a+b/x)^(3/2)/(c+d/x),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ {\left (\frac {2 \, d^{3} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{3} c^{3} - a b^{2} c^{2} d\right )} \sqrt {b c d - a d^{2}}} + \frac {2 \, a b c - \frac {3 \, {\left (a x + b\right )} b c}{x} + \frac {{\left (a x + b\right )} a d}{x}}{{\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} {\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}} + \frac {{\left (3 \, b c + 2 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} b^{2} c^{2}}\right )} b^{2} \]