\[ \int x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {3 \left (-a d +b c \right ) \left (-a d +5 b c \right ) e^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {d}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{\sqrt {b}\, \sqrt {e}}\right )}{8 d^{\frac {7}{2}} \sqrt {b}}-\frac {c \left (-a d +b c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{d^{3}}-\frac {\left (-5 a d +9 b c \right ) e \left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{8 d^{3}}+\frac {b e \left (d \,x^{2}+c \right )^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{4 d^{3}} \]
command
integrate(x^3*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{16} \, {\left (2 \, \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c} {\left (\frac {2 \, b x^{2} \mathrm {sgn}\left (d x^{2} + c\right )}{d^{2}} - \frac {7 \, b^{2} c d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) - 5 \, a b d^{6} \mathrm {sgn}\left (d x^{2} + c\right )}{b d^{8}}\right )} - \frac {16 \, {\left (b^{2} c^{3} \mathrm {sgn}\left (d x^{2} + c\right ) - 2 \, a b c^{2} d \mathrm {sgn}\left (d x^{2} + c\right ) + a^{2} c d^{2} \mathrm {sgn}\left (d x^{2} + c\right )\right )}}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} d + \sqrt {b d} c\right )} d^{3}} - \frac {3 \, {\left (5 \, \sqrt {b d} b^{2} c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) - 6 \, \sqrt {b d} a b c d \mathrm {sgn}\left (d x^{2} + c\right ) + \sqrt {b d} a^{2} d^{2} \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} b d - \sqrt {b d} b c - \sqrt {b d} a d \right |}\right )}{b d^{4}}\right )} e^{\frac {3}{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________