\[ \int x^2 \text {ArcCos}\left (a x^2\right ) \, dx \]
Optimal antiderivative \[ \frac {x^{3} \arccos \left (a \,x^{2}\right )}{3}+\frac {2 \EllipticF \left (x \sqrt {a}, i\right )}{9 a^{\frac {3}{2}}}-\frac {2 x \sqrt {-a^{2} x^{4}+1}}{9 a} \]
command
integrate(x^2*arccos(a*x^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {3 \, a x^{3} \arccos \left (a x^{2}\right ) - 2 \, \sqrt {-a^{2} x^{4} + 1} x}{9 \, a} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{2} \arccos \left (a x^{2}\right ), x\right ) \]