\[ \int \frac {x \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {e \sqrt {e \,x^{2}+d}}{c}-\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e \,x^{2}+d}}{\sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \left (2 c^{2} d^{2}+b \,e^{2} \left (b -\sqrt {-4 a c +b^{2}}\right )-2 c e \left (b d +a e -d \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 c^{\frac {3}{2}} \sqrt {-4 a c +b^{2}}\, \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}+\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e \,x^{2}+d}}{\sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \left (2 c^{2} d^{2}+b \,e^{2} \left (b +\sqrt {-4 a c +b^{2}}\right )-2 c e \left (b d +a e +d \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 c^{\frac {3}{2}} \sqrt {-4 a c +b^{2}}\, \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate(x*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]