\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^3 \, dx \]
Optimal antiderivative \[ -\frac {34 a^{3} \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{99 d e}+\frac {34 a^{3} e \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{77 d}-\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}} \left (a +a \sin \left (d x +c \right )\right )^{2}}{13 d e}-\frac {34 \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}} \left (a^{3}+a^{3} \sin \left (d x +c \right )\right )}{143 d e}+\frac {170 a^{3} e^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {e \cos \left (d x +c \right )}}+\frac {170 a^{3} e^{3} \sin \left (d x +c \right ) \sqrt {e \cos \left (d x +c \right )}}{231 d} \]
command
integrate((e*cos(d*x+c))^(7/2)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-3315 i \, \sqrt {2} a^{3} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 3315 i \, \sqrt {2} a^{3} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (693 \, a^{3} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} - 4004 \, a^{3} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 39 \, {\left (63 \, a^{3} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 51 \, a^{3} \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 85 \, a^{3} e^{\frac {7}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{9009 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (3 \, a^{3} e^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} e^{3} \cos \left (d x + c\right )^{3} + {\left (a^{3} e^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} e^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]