\[ \int \frac {\left (c+\frac {d}{x}\right )^3}{\sqrt {a+\frac {b}{x}}} \, dx \]
Optimal antiderivative \[ -\frac {c^{2} \left (-6 a d +b c \right ) \arctanh \! \left (\frac {\sqrt {a +\frac {b}{x}}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {d \left (-4 a^{2} d^{2}+18 a b c d +6 b^{2} c^{2}+\frac {b d \left (2 a d +3 b c \right )}{x}\right ) \sqrt {a +\frac {b}{x}}}{3 a \,b^{2}}+\frac {c \left (c +\frac {d}{x}\right )^{2} x \sqrt {a +\frac {b}{x}}}{a} \]
command
integrate((c+d/x)^3/(a+b/x)^(1/2),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
\[ -\frac {\frac {3 \, b^{2} c^{3} \sqrt {\frac {a x + b}{x}}}{{\left (a - \frac {a x + b}{x}\right )} a} - \frac {3 \, {\left (b^{2} c^{3} - 6 \, a b c^{2} d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2 \, {\left (9 \, b^{3} c d^{2} \sqrt {\frac {a x + b}{x}} - 3 \, a b^{2} d^{3} \sqrt {\frac {a x + b}{x}} + \frac {{\left (a x + b\right )} b^{2} d^{3} \sqrt {\frac {a x + b}{x}}}{x}\right )}}{b^{3}}}{3 \, b} \]