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