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