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