\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx \]
Optimal antiderivative \[ -\frac {34 a b \left (53 a^{2}+38 b^{2}\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{6435 d e}+\frac {2 \left (55 a^{4}+60 a^{2} b^{2}+4 b^{4}\right ) e \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{385 d}-\frac {2 b \left (93 a^{2}+26 b^{2}\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}} \left (a +b \sin \left (d x +c \right )\right )}{715 d e}-\frac {14 a b \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}} \left (a +b \sin \left (d x +c \right )\right )^{2}}{65 d e}-\frac {2 b \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}} \left (a +b \sin \left (d x +c \right )\right )^{3}}{15 d e}+\frac {2 \left (55 a^{4}+60 a^{2} b^{2}+4 b^{4}\right ) e^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {e \cos \left (d x +c \right )}}+\frac {2 \left (55 a^{4}+60 a^{2} b^{2}+4 b^{4}\right ) e^{3} \sin \left (d x +c \right ) \sqrt {e \cos \left (d x +c \right )}}{231 d} \]
command
integrate((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-195 i \, \sqrt {2} {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (13860 \, a b^{3} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} - 20020 \, {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} + 39 \, {\left (77 \, b^{4} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} - 7 \, {\left (90 \, a^{2} b^{2} + 17 \, b^{4}\right )} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} + 3 \, {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} + 5 \, {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{45045 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{4} e^{3} \cos \left (d x + c\right )^{7} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} e^{3} \cos \left (d x + c\right )^{5} + {\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} e^{3} \cos \left (d x + c\right )^{3} - 4 \, {\left (a b^{3} e^{3} \cos \left (d x + c\right )^{5} - {\left (a^{3} b + a b^{3}\right )} e^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]