\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx \]
Optimal antiderivative \[ \frac {3 d^{\frac {5}{2}} \left (8 a^{2} d^{2}-24 a b c d +21 b^{2} c^{2}\right ) \arctan \! \left (\frac {\sqrt {d}\, \sqrt {a +\frac {b}{x}}}{\sqrt {-a d +b c}}\right )}{4 c^{4} \left (-a d +b c \right )^{\frac {7}{2}}}-\frac {3 \left (2 a d +b c \right ) \arctanh \! \left (\frac {\sqrt {a +\frac {b}{x}}}{\sqrt {a}}\right )}{a^{\frac {5}{2}} c^{4}}+\frac {3 b \left (-a d +2 b c \right ) \left (4 a^{2} d^{2}-a b c d +2 b^{2} c^{2}\right )}{4 a^{2} c^{3} \left (-a d +b c \right )^{3} \sqrt {a +\frac {b}{x}}}+\frac {d \left (-3 a d +2 b c \right )}{2 a \,c^{2} \left (-a d +b c \right ) \left (c +\frac {d}{x}\right )^{2} \sqrt {a +\frac {b}{x}}}+\frac {d \left (12 a^{2} d^{2}-21 a b c d +4 b^{2} c^{2}\right )}{4 a \,c^{3} \left (-a d +b c \right )^{2} \left (c +\frac {d}{x}\right ) \sqrt {a +\frac {b}{x}}}+\frac {x}{a c \left (c +\frac {d}{x}\right )^{2} \sqrt {a +\frac {b}{x}}} \]
command
integrate(1/(a+b/x)^(3/2)/(c+d/x)^3,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
\[ \frac {1}{4} \, b^{4} {\left (\frac {3 \, {\left (21 \, b^{2} c^{2} d^{3} - 24 \, a b c d^{4} + 8 \, a^{2} d^{5}\right )} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{7} c^{7} - 3 \, a b^{6} c^{6} d + 3 \, a^{2} b^{5} c^{5} d^{2} - a^{3} b^{4} c^{4} d^{3}\right )} \sqrt {b c d - a d^{2}}} + \frac {4 \, {\left (2 \, a b^{3} c^{3} - \frac {3 \, {\left (a x + b\right )} b^{3} c^{3}}{x} + \frac {3 \, {\left (a x + b\right )} a b^{2} c^{2} d}{x} - \frac {3 \, {\left (a x + b\right )} a^{2} b c d^{2}}{x} + \frac {{\left (a x + b\right )} a^{3} d^{3}}{x}\right )}}{{\left (a^{2} b^{6} c^{6} - 3 \, a^{3} b^{5} c^{5} d + 3 \, a^{4} b^{4} c^{4} d^{2} - a^{5} b^{3} c^{3} d^{3}\right )} {\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}} + \frac {17 \, b^{2} c^{2} d^{3} \sqrt {\frac {a x + b}{x}} - 25 \, a b c d^{4} \sqrt {\frac {a x + b}{x}} + 8 \, a^{2} d^{5} \sqrt {\frac {a x + b}{x}} + \frac {15 \, {\left (a x + b\right )} b c d^{4} \sqrt {\frac {a x + b}{x}}}{x} - \frac {8 \, {\left (a x + b\right )} a d^{5} \sqrt {\frac {a x + b}{x}}}{x}}{{\left (b^{6} c^{6} - 3 \, a b^{5} c^{5} d + 3 \, a^{2} b^{4} c^{4} d^{2} - a^{3} b^{3} c^{3} d^{3}\right )} {\left (b c - a d + \frac {{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac {12 \, {\left (b c + 2 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} b^{4} c^{4}}\right )} \]