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