\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {x \left (b \,x^{2}+a \right )}{8 a \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}+\frac {7 x \left (b \,x^{2}+a \right )^{2}}{48 a^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}+\frac {35 x \left (b \,x^{2}+a \right )^{3}}{192 a^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}+\frac {35 x \left (b \,x^{2}+a \right )^{4}}{128 a^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}+\frac {35 \left (b \,x^{2}+a \right )^{5} \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}} \sqrt {b}} \]
command
integrate(1/(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
\[ \frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {105 \, b^{3} x^{7} + 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} + 279 \, a^{3} x}{384 \, {\left (b x^{2} + a\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________