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