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