\[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 b \arctanh \left (\frac {2 a +\frac {b}{x}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}\right )}{2 a^{\frac {5}{2}}}-\frac {2 \left (b^{2}-2 a c +\frac {b c}{x}\right ) x}{a \left (-4 a c +b^{2}\right ) \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}+\frac {\left (-8 a c +3 b^{2}\right ) x \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}{a^{2} \left (-4 a c +b^{2}\right )} \]
command
integrate(1/(a+c/x^2+b/x)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (3 \, b^{3} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 12 \, a b c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{2} \sqrt {c} - 16 \, a^{\frac {3}{2}} c^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{2 \, {\left (a^{\frac {5}{2}} b^{2} - 4 \, a^{\frac {7}{2}} c\right )}} + \frac {{\left (\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} x}{a^{2} b^{2} \mathrm {sgn}\left (x\right ) - 4 \, a^{3} c \mathrm {sgn}\left (x\right )} + \frac {3 \, b^{3} - 10 \, a b c}{a^{2} b^{2} \mathrm {sgn}\left (x\right ) - 4 \, a^{3} c \mathrm {sgn}\left (x\right )}\right )} x + \frac {3 \, b^{2} c - 8 \, a c^{2}}{a^{2} b^{2} \mathrm {sgn}\left (x\right ) - 4 \, a^{3} c \mathrm {sgn}\left (x\right )}}{\sqrt {a x^{2} + b x + c}} + \frac {3 \, b \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x + c}\right )} \sqrt {a} - b \right |}\right )}{2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________