\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {x}{3 d \left (-b e +2 c d \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {c^{2} \arctanh \left (\frac {x \sqrt {e}\, \sqrt {-b e +2 c d}}{\sqrt {-b e +c d}\, \sqrt {e \,x^{2}+d}}\right )}{\left (-b e +2 c d \right )^{\frac {5}{2}} \sqrt {e}\, \sqrt {-b e +c d}}-\frac {\left (-2 b e +7 c d \right ) x}{3 d^{2} \left (-b e +2 c d \right )^{2} \sqrt {e \,x^{2}+d}} \]
command
integrate(1/(e*x^2+d)^(3/2)/(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
\[ -\frac {c^{2} \arctan \left (\frac {{\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} c - 3 \, c d + 2 \, b e}{2 \, \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}}\right ) e^{\frac {1}{2}}}{{\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}} - \frac {{\left (\frac {{\left (28 \, c^{3} d^{3} e^{2} - 36 \, b c^{2} d^{2} e^{3} + 15 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{2}}{16 \, c^{4} d^{6} e - 32 \, b c^{3} d^{5} e^{2} + 24 \, b^{2} c^{2} d^{4} e^{3} - 8 \, b^{3} c d^{3} e^{4} + b^{4} d^{2} e^{5}} + \frac {3 \, {\left (12 \, c^{3} d^{4} e - 16 \, b c^{2} d^{3} e^{2} + 7 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )}}{16 \, c^{4} d^{6} e - 32 \, b c^{3} d^{5} e^{2} + 24 \, b^{2} c^{2} d^{4} e^{3} - 8 \, b^{3} c d^{3} e^{4} + b^{4} d^{2} e^{5}}\right )} x}{3 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________