\[ \int \frac {x^2 \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-5 a e +b d \right ) x}{8 a \,b^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {\left (-a e +b d \right ) x}{4 b^{2} \left (b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {\left (3 a e +b d \right ) \left (b \,x^{2}+a \right ) \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}} b^{\frac {5}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}} \]
command
integrate(x^2*(e*x^2+d)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (b d + 3 \, a e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {b^{2} d x^{3} - 5 \, a b x^{3} e - a b d x - 3 \, a^{2} x e}{8 \, {\left (b x^{2} + a\right )}^{2} a b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________