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