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