\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 a \left (3 a^{2}+2 b^{2}\right ) e \cos \left (d x +c \right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{15 d}+\frac {2 b \left (43 a^{2}+12 b^{2}\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{231 d e}+\frac {10 a b \left (a +b \cos \left (d x +c \right )\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{33 d e}+\frac {2 b \left (a +b \cos \left (d x +c \right )\right )^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{11 d e}-\frac {2 a \left (3 a^{2}+2 b^{2}\right ) e^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \sin \left (d x +c \right )}}{5 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {\sin \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {231 i \, \sqrt {2} \sqrt {-i} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} \sqrt {i} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (105 \, b^{3} \cos \left (d x + c\right )^{4} e^{\frac {5}{2}} + 385 \, a b^{2} \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} + 45 \, {\left (11 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} + 231 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) e^{\frac {5}{2}} - 15 \, {\left (33 \, a^{2} b + 4 \, b^{3}\right )} e^{\frac {5}{2}}\right )} \sin \left (d x + c\right )^{\frac {3}{2}}}{1155 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b^{3} e^{2} \cos \left (d x + c\right )^{5} + 3 \, a b^{2} e^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} b e^{2} \cos \left (d x + c\right ) + {\left (3 \, a^{2} b - b^{3}\right )} e^{2} \cos \left (d x + c\right )^{3} - a^{3} e^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} e^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {e \sin \left (d x + c\right )}, x\right ) \]