\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {x}{4 d \left (-b e +2 c d \right ) \left (e \,x^{2}+d \right )^{2}}-\frac {\left (-3 b e +10 c d \right ) x}{8 d^{2} \left (-b e +2 c d \right )^{2} \left (e \,x^{2}+d \right )}-\frac {\left (3 b^{2} e^{2}-16 b c d e +28 c^{2} d^{2}\right ) \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}} \left (-b e +2 c d \right )^{3} \sqrt {e}}-\frac {c^{\frac {5}{2}} \arctanh \left (\frac {x \sqrt {c}\, \sqrt {e}}{\sqrt {-b e +c d}}\right )}{\left (-b e +2 c d \right )^{3} \sqrt {e}\, \sqrt {-b e +c d}} \]
command
integrate(1/(e*x^2+d)^2/(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
\[ \frac {c^{3} \arctan \left (\frac {c x e}{\sqrt {-c^{2} d e + b c e^{2}}}\right )}{{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {-c^{2} d e + b c e^{2}}} - \frac {{\left (28 \, c^{2} d^{2} - 16 \, b c d e + 3 \, b^{2} e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{8 \, {\left (8 \, c^{3} d^{5} - 12 \, b c^{2} d^{4} e + 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \sqrt {d}} - \frac {10 \, c d x^{3} e - 3 \, b x^{3} e^{2} + 14 \, c d^{2} x - 5 \, b d x e}{8 \, {\left (4 \, c^{2} d^{4} - 4 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (x^{2} e + d\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________