\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{6 d^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {x \sqrt {2}\, \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right ) \sqrt {2}}{8 d^{3} \sqrt {e}}+\frac {7 x}{12 d^{3} \sqrt {e \,x^{2}+d}} \]
command
integrate(1/(e*x^2+d)^(3/2)/(-e^2*x^4+d^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {x {\left (\frac {7 \, x^{2} e}{d^{3}} + \frac {9}{d^{2}}\right )}}{12 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}}} + \frac {\sqrt {2} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {{\left | 2 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{16 \, d^{2} {\left | d \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________