\[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b^{2}-2 a c +\frac {b c}{x}\right ) x}{3 a \left (-4 a c +b^{2}\right ) \left (a +\frac {c}{x^{2}}+\frac {b}{x}\right )^{\frac {3}{2}}}-\frac {5 b \arctanh \left (\frac {2 a +\frac {b}{x}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}\right )}{2 a^{\frac {7}{2}}}-\frac {2 \left (5 b^{4}-32 a \,b^{2} c +32 a^{2} c^{2}+\frac {b c \left (-28 a c +5 b^{2}\right )}{x}\right ) x}{3 a^{2} \left (-4 a c +b^{2}\right )^{2} \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}+\frac {\left (128 a^{2} c^{2}-100 a \,b^{2} c +15 b^{4}\right ) x \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}{3 a^{3} \left (-4 a c +b^{2}\right )^{2}} \]
command
integrate(1/(a+c/x^2+b/x)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (15 \, b^{5} \sqrt {c} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 120 \, a b^{3} c^{\frac {3}{2}} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 240 \, a^{2} b c^{\frac {5}{2}} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{4} c - 200 \, a^{\frac {3}{2}} b^{2} c^{2} + 256 \, a^{\frac {5}{2}} c^{3}\right )} \mathrm {sgn}\left (x\right )}{6 \, {\left (a^{\frac {7}{2}} b^{4} \sqrt {c} - 8 \, a^{\frac {9}{2}} b^{2} c^{\frac {3}{2}} + 16 \, a^{\frac {11}{2}} c^{\frac {5}{2}}\right )}} + \frac {{\left ({\left ({\left (\frac {3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} x}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )} + \frac {4 \, {\left (5 \, a b^{5} - 37 \, a^{2} b^{3} c + 64 \, a^{3} b c^{2}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {3 \, {\left (5 \, b^{6} - 30 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {6 \, {\left (5 \, b^{5} c - 35 \, a b^{3} c^{2} + 52 \, a^{2} b c^{3}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}}{3 \, {\left (a x^{2} + b x + c\right )}^{\frac {3}{2}}} + \frac {5 \, 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 {7}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________