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