\[ \int (c e+d e x)^{5/2} (a+b \text {ArcSin}(c+d x)) \, dx \]
Optimal antiderivative \[ \frac {2 \left (e \left (d x +c \right )\right )^{\frac {7}{2}} \left (a +b \arcsin \left (d x +c \right )\right )}{7 d e}-\frac {20 b \,e^{\frac {5}{2}} \EllipticF \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}, i\right )}{147 d}+\frac {4 b \left (e \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {1-\left (d x +c \right )^{2}}}{49 d}+\frac {20 b \,e^{2} \sqrt {e \left (d x +c \right )}\, \sqrt {1-\left (d x +c \right )^{2}}}{147 d} \]
command
integrate((d*e*x+c*e)^(5/2)*(a+b*arcsin(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (10 \, \sqrt {-d^{3} e} b e^{2} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (21 \, {\left (b d^{5} x^{3} + 3 \, b c d^{4} x^{2} + 3 \, b c^{2} d^{3} x + b c^{3} d^{2}\right )} \arcsin \left (d x + c\right ) e^{2} + 2 \, {\left (3 \, b d^{4} x^{2} + 6 \, b c d^{3} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} e^{2} + 21 \, {\left (a d^{5} x^{3} + 3 \, a c d^{4} x^{2} + 3 \, a c^{2} d^{3} x + a c^{3} d^{2}\right )} e^{2}\right )} \sqrt {d x + c} e^{\frac {1}{2}}\right )}}{147 \, d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a d^{2} e^{2} x^{2} + 2 \, a c d e^{2} x + a c^{2} e^{2} + {\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + b c^{2} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]