\[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {x}{2 d \left (-b e +2 c d \right ) \left (e \,x^{2}+d \right )}-\frac {\left (-b e +4 c d \right ) \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}} \left (-b e +2 c d \right )^{2} \sqrt {e}}-\frac {c^{\frac {3}{2}} \arctanh \left (\frac {x \sqrt {c}\, \sqrt {e}}{\sqrt {-b e +c d}}\right )}{\left (-b e +2 c d \right )^{2} \sqrt {e}\, \sqrt {-b e +c d}} \]
command
integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {c^{2} \arctan \left (\frac {c x e}{\sqrt {-c^{2} d e + b c e^{2}}}\right )}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {-c^{2} d e + b c e^{2}}} - \frac {{\left (4 \, c d - b e\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, {\left (4 \, c^{2} d^{3} - 4 \, b c d^{2} e + b^{2} d e^{2}\right )} \sqrt {d}} - \frac {x}{2 \, {\left (2 \, c d^{2} - b d e\right )} {\left (x^{2} e + d\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________