\[ \int x^4 \left (a+b \text {ArcSin}\left (c x^2\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {x^{5} \left (a +b \arcsin \left (c \,x^{2}\right )\right )}{5}-\frac {6 b \EllipticE \left (x \sqrt {c}, i\right )}{25 c^{\frac {5}{2}}}+\frac {6 b \EllipticF \left (x \sqrt {c}, i\right )}{25 c^{\frac {5}{2}}}+\frac {2 b \,x^{3} \sqrt {-c^{2} x^{4}+1}}{25 c} \]
command
integrate(x^4*(a+b*arcsin(c*x^2)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 \, b c^{3} x^{6} \arcsin \left (c x^{2}\right ) + 5 \, a c^{3} x^{6} + 2 \, {\left (b c^{2} x^{4} + 3 \, b\right )} \sqrt {-c^{2} x^{4} + 1}}{25 \, c^{3} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (b x^{4} \arcsin \left (c x^{2}\right ) + a x^{4}, x\right ) \]