\[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ -\frac {2 a e \cos \left (d x +c \right ) \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{7 d}+\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d e}-\frac {10 a \,e^{4} \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\sin }\left (d x +c \right )\right )}{21 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {e \sin \left (d x +c \right )}}-\frac {10 a \,e^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}{21 d} \]
command
integrate((a+b*cos(d*x+c))*(e*sin(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {15 \, \sqrt {2} \sqrt {-i} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} \sqrt {i} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (7 \, b \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} + 9 \, a \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} - 14 \, b \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 24 \, a \cos \left (d x + c\right ) e^{\frac {7}{2}} + 7 \, b e^{\frac {7}{2}}\right )} \sqrt {\sin \left (d x + c\right )}}{63 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b e^{3} \cos \left (d x + c\right )^{3} + a e^{3} \cos \left (d x + c\right )^{2} - b e^{3} \cos \left (d x + c\right ) - a e^{3}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ), x\right ) \]