\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx \]
Optimal antiderivative \[ -\frac {2 b \left (177 a^{2}+44 b^{2}\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{1287 d e}+\frac {2 a \left (11 a^{2}+6 b^{2}\right ) e \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{77 d}-\frac {34 a b \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}} \left (a +b \sin \left (d x +c \right )\right )}{143 d e}-\frac {2 b \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}} \left (a +b \sin \left (d x +c \right )\right )^{2}}{13 d e}+\frac {10 a \left (11 a^{2}+6 b^{2}\right ) e^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {e \cos \left (d x +c \right )}}+\frac {10 a \left (11 a^{2}+6 b^{2}\right ) e^{3} \sin \left (d x +c \right ) \sqrt {e \cos \left (d x +c \right )}}{231 d} \]
command
integrate((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-195 i \, \sqrt {2} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (693 \, b^{3} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} - 1001 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 39 \, {\left (63 \, a b^{2} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 3 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 5 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{9009 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (3 \, a b^{2} e^{3} \cos \left (d x + c\right )^{5} - {\left (a^{3} + 3 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right )^{3} + {\left (b^{3} e^{3} \cos \left (d x + c\right )^{5} - {\left (3 \, a^{2} b + b^{3}\right )} e^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]