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