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