\[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 c e}-\frac {b \sqrt {e \,x^{2}+d}}{c^{2}}+\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e \,x^{2}+d}}{\sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \left (b c d -b^{2} e +a c e +\frac {-3 a b c e +2 a \,c^{2} d +b^{3} e -b^{2} c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 c^{\frac {5}{2}} \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}+\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e \,x^{2}+d}}{\sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \left (b c d -b^{2} e +a c e +\frac {3 a b c e -2 a \,c^{2} d -b^{3} e +b^{2} c d}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {2}}{2 c^{\frac {5}{2}} \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate(x^5*(e*x^2+d)^(1/2)/(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 \[ \text {Timed out} \]