\[ \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {c \arctanh \left (\frac {x \sqrt {e}\, \sqrt {-b e +2 c d}}{\sqrt {-b e +c d}\, \sqrt {e \,x^{2}+d}}\right )}{\left (-b e +2 c d \right )^{\frac {3}{2}} \sqrt {e}\, \sqrt {-b e +c d}}-\frac {x}{d \left (-b e +2 c d \right ) \sqrt {e \,x^{2}+d}} \]
command
integrate(1/(e*x^2+d)^(1/2)/(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
\[ -\frac {c \arctan \left (\frac {{\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} c - 3 \, c d + 2 \, b e}{2 \, \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}}\right ) e^{\frac {1}{2}}}{\sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}} {\left (2 \, c d e - b e^{2}\right )}} - \frac {x}{{\left (2 \, c d^{2} - b d e\right )} \sqrt {x^{2} e + d}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________