\[ \int \frac {1}{x^5 \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{b \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {8 \sqrt {c \,x^{4}+b \,x^{2}}}{7 b^{2} x^{8}}+\frac {48 c \sqrt {c \,x^{4}+b \,x^{2}}}{35 b^{3} x^{6}}-\frac {64 c^{2} \sqrt {c \,x^{4}+b \,x^{2}}}{35 b^{4} x^{4}}+\frac {128 c^{3} \sqrt {c \,x^{4}+b \,x^{2}}}{35 b^{5} x^{2}} \]
command
integrate(1/x^5/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {c^{4} x}{\sqrt {c x^{2} + b} b^{5} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} c^{\frac {7}{2}} - 280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} b c^{\frac {7}{2}} + 1015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} b^{2} c^{\frac {7}{2}} - 2240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} b^{3} c^{\frac {7}{2}} + 1673 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{4} c^{\frac {7}{2}} - 616 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{5} c^{\frac {7}{2}} + 93 \, b^{6} c^{\frac {7}{2}}\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{7} b^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{5}}\,{d x} \]________________________________________________________________________________________