\[ \int \frac {a+b \text {ArcSin}(c+d x)}{(c e+d e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{5 d e \left (e \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {4 b \EllipticF \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}, i\right )}{15 d \,e^{\frac {7}{2}}}-\frac {4 b \sqrt {1-\left (d x +c \right )^{2}}}{15 d \,e^{2} \left (e \left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (3 \, b d^{2} \arcsin \left (d x + c\right ) + 3 \, a d^{2} + 2 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d x + c} e^{\frac {1}{2}} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )} e^{\left (-4\right )}}{15 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]