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