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