\[ \int x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{3}}{6 b \,d^{2} e}-\frac {\left (-a d +b c \right ) \left (-a^{2} d^{2}-10 a b c d +35 b^{2} c^{2}\right ) e^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {d}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{\sqrt {b}\, \sqrt {e}}\right )}{16 b^{\frac {3}{2}} d^{\frac {9}{2}}}+\frac {c^{2} \left (-a d +b c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{d^{4}}+\frac {\left (-5 a^{2} d^{2}-50 a b c d +79 b^{2} c^{2}\right ) e \left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 b \,d^{4}}-\frac {\left (a d +11 b c \right ) e \left (d \,x^{2}+c \right )^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{24 d^{4}} \]
command
integrate(x^5*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{96} \, {\left (2 \, \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c} {\left (2 \, {\left (\frac {4 \, b x^{2} \mathrm {sgn}\left (d x^{2} + c\right )}{d^{2}} - \frac {11 \, b^{3} c d^{10} \mathrm {sgn}\left (d x^{2} + c\right ) - 7 \, a b^{2} d^{11} \mathrm {sgn}\left (d x^{2} + c\right )}{b^{2} d^{13}}\right )} x^{2} + \frac {57 \, b^{3} c^{2} d^{9} \mathrm {sgn}\left (d x^{2} + c\right ) - 52 \, a b^{2} c d^{10} \mathrm {sgn}\left (d x^{2} + c\right ) + 3 \, a^{2} b d^{11} \mathrm {sgn}\left (d x^{2} + c\right )}{b^{2} d^{13}}\right )} + \frac {96 \, {\left (b^{2} c^{4} \mathrm {sgn}\left (d x^{2} + c\right ) - 2 \, a b c^{3} d \mathrm {sgn}\left (d x^{2} + c\right ) + a^{2} c^{2} d^{2} \mathrm {sgn}\left (d x^{2} + c\right )\right )}}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} d + \sqrt {b d} c\right )} d^{4}} + \frac {3 \, {\left (35 \, \sqrt {b d} b^{3} c^{3} \mathrm {sgn}\left (d x^{2} + c\right ) - 45 \, \sqrt {b d} a b^{2} c^{2} d \mathrm {sgn}\left (d x^{2} + c\right ) + 9 \, \sqrt {b d} a^{2} b c d^{2} \mathrm {sgn}\left (d x^{2} + c\right ) + \sqrt {b d} a^{3} d^{3} \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} b d - \sqrt {b d} b c - \sqrt {b d} a d \right |}\right )}{b^{2} d^{5}}\right )} e^{\frac {3}{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________