\[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ \frac {d^{2} x^{3} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{3}+\frac {2 d e \,x^{5} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{5}+\frac {e^{2} x^{7} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{7}-\frac {b \left (280 c^{4} d^{2}+252 c^{2} d e +75 e^{2}\right ) x \arctanh \left (\frac {c x}{\sqrt {c^{2} x^{2}-1}}\right )}{1680 c^{6} \sqrt {c^{2} x^{2}}}-\frac {b \left (280 c^{4} d^{2}+252 c^{2} d e +75 e^{2}\right ) x^{2} \sqrt {c^{2} x^{2}-1}}{1680 c^{5} \sqrt {c^{2} x^{2}}}-\frac {b e \left (84 c^{2} d +25 e \right ) x^{4} \sqrt {c^{2} x^{2}-1}}{840 c^{3} \sqrt {c^{2} x^{2}}}-\frac {b \,e^{2} x^{6} \sqrt {c^{2} x^{2}-1}}{42 c \sqrt {c^{2} x^{2}}} \]
command
integrate(x^2*(e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________