\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )} \, dx \]
Optimal antiderivative \[ \frac {b \left (-3 a d +5 b c \right )}{3 a^{2} c \left (-a d +b c \right ) \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}+\frac {x}{a c \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}-\frac {2 d^{\frac {7}{2}} \arctan \! \left (\frac {\sqrt {d}\, \sqrt {a +\frac {b}{x}}}{\sqrt {-a d +b c}}\right )}{c^{2} \left (-a d +b c \right )^{\frac {5}{2}}}-\frac {\left (2 a d +5 b c \right ) \arctanh \! \left (\frac {\sqrt {a +\frac {b}{x}}}{\sqrt {a}}\right )}{a^{\frac {7}{2}} c^{2}}+\frac {b \left (a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right )}{a^{3} c \left (-a d +b c \right )^{2} \sqrt {a +\frac {b}{x}}} \]
command
integrate(1/(a+b/x)^(5/2)/(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
\[ -\frac {1}{3} \, {\left (\frac {6 \, d^{4} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} \sqrt {b c d - a d^{2}}} - \frac {2 \, {\left (a b c - a^{2} d + \frac {6 \, {\left (a x + b\right )} b c}{x} - \frac {9 \, {\left (a x + b\right )} a d}{x}\right )} x}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} {\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}} + \frac {3 \, \sqrt {\frac {a x + b}{x}}}{{\left (a - \frac {a x + b}{x}\right )} a^{3} b c} - \frac {3 \, {\left (5 \, b c + 2 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3} b^{2} c^{2}}\right )} b^{2} \]