\[ \int \frac {x^5}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right ) \left (5 a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {d}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{\sqrt {b}\, \sqrt {e}}\right )}{16 b^{\frac {7}{2}} d^{\frac {5}{2}} \sqrt {e}}+\frac {\left (5 a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{16 b^{3} d^{2} e}-\frac {\left (5 a d +3 b c \right ) \left (d \,x^{2}+c \right )^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{24 b^{2} d^{2} e}-\frac {\left (d \,x^{2}+c \right )^{3} \left (a -\frac {c \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{6 b d \left (-a d +b c \right ) e} \]
command
integrate(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
\[ \frac {{\left (2 \, \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{b} + \frac {b^{2} c d - 5 \, a b d^{2}}{b^{3} d^{2}}\right )} - \frac {3 \, b^{2} c^{2} + 4 \, a b c d - 15 \, a^{2} d^{2}}{b^{3} d^{2}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -b c - a d - 2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} \sqrt {b d} \right |}\right )}{\sqrt {b d} b^{3} d^{2}}\right )} e^{\left (-\frac {1}{2}\right )}}{96 \, \mathrm {sgn}\left (d x^{2} + c\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________