\[ \int (d x)^{3/2} (a+b \text {ArcSin}(c x)) \, dx \]
Optimal antiderivative \[ \frac {2 \left (d x \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}{5 d}-\frac {12 b \,d^{\frac {3}{2}} \EllipticE \left (\frac {\sqrt {c}\, \sqrt {d x}}{\sqrt {d}}, i\right )}{25 c^{\frac {5}{2}}}+\frac {12 b \,d^{\frac {3}{2}} \EllipticF \left (\frac {\sqrt {c}\, \sqrt {d x}}{\sqrt {d}}, i\right )}{25 c^{\frac {5}{2}}}+\frac {4 b \left (d x \right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}{25 c} \]
command
integrate((d*x)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (6 \, \sqrt {-c^{2} d} b d {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) - {\left (5 \, b c^{3} d x^{2} \arcsin \left (c x\right ) + 5 \, a c^{3} d x^{2} + 2 \, \sqrt {-c^{2} x^{2} + 1} b c^{2} d x\right )} \sqrt {d x}\right )}}{25 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b d x \arcsin \left (c x\right ) + a d x\right )} \sqrt {d x}, x\right ) \]