\[ \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {231}{128 a^{4} x^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {1}{8 a \,x^{3} \left (b \,x^{2}+a \right )^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {11}{48 a^{2} x^{3} \left (b \,x^{2}+a \right )^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {33}{64 a^{3} x^{3} \left (b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {385 \left (b \,x^{2}+a \right )}{128 a^{5} x^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {1155 b \left (b \,x^{2}+a \right )}{128 a^{6} x \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {1155 b^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{128 a^{\frac {13}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}} \]
command
integrate(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1155 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {15 \, b x^{2} - a}{3 \, a^{6} x^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1545 \, b^{5} x^{7} + 5153 \, a b^{4} x^{5} + 5855 \, a^{2} b^{3} x^{3} + 2295 \, a^{3} b^{2} x}{384 \, {\left (b x^{2} + a\right )}^{4} a^{6} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________