\[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a d +b c \right ) \left (a d +3 b c \right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{\sqrt {a}\, \sqrt {e}}\right )}{8 a^{\frac {5}{2}} c^{\frac {3}{2}} \sqrt {e}}-\frac {\left (-a d +b c \right )^{2} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{4 a c \left (a e -\frac {c e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{2}}-\frac {\left (-a d +b c \right ) \left (a d +3 b c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{8 a^{2} c \left (a e -\frac {c e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )} \]
command
integrate(1/x^5/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a^{2} c} - \frac {3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} b^{2} c^{2} - 5 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a b^{2} c^{3} - 2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a b c d - 10 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{2} b c^{2} d - {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{2} d^{2} - {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{3} c d^{2} - 8 \, \sqrt {b d} a^{2} b c^{3} - 8 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} \sqrt {b d} a^{2} c d}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} - a c\right )}^{2} a^{2} c}\right )} e^{\left (-\frac {1}{2}\right )}}{8 \, \mathrm {sgn}\left (d x^{2} + c\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________