\[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ \frac {d^{2} x^{4} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{4}+\frac {d e \,x^{6} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{3}+\frac {e^{2} x^{8} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{8}-\frac {b \left (6 c^{4} d^{2}+16 c^{2} d e +9 e^{2}\right ) x \left (c^{2} x^{2}-1\right )^{\frac {3}{2}}}{72 c^{7} \sqrt {c^{2} x^{2}}}-\frac {b e \left (8 c^{2} d +9 e \right ) x \left (c^{2} x^{2}-1\right )^{\frac {5}{2}}}{120 c^{7} \sqrt {c^{2} x^{2}}}-\frac {b \,e^{2} x \left (c^{2} x^{2}-1\right )^{\frac {7}{2}}}{56 c^{7} \sqrt {c^{2} x^{2}}}-\frac {b \left (6 c^{4} d^{2}+8 c^{2} d e +3 e^{2}\right ) x \sqrt {c^{2} x^{2}-1}}{24 c^{7} \sqrt {c^{2} x^{2}}} \]
command
integrate(x^3*(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 \[ \text {Timed out} \]_______________________________________________________