\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (89 a^{2}+28 b^{2}\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{315 d e}+\frac {26 a b \left (a +b \cos \left (d x +c \right )\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{63 d e}+\frac {2 b \left (a +b \cos \left (d x +c \right )\right )^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{9 d e}-\frac {2 a \left (7 a^{2}+6 b^{2}\right ) e^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\sin }\left (d x +c \right )\right )}{21 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {e \sin \left (d x +c \right )}}-\frac {2 a \left (7 a^{2}+6 b^{2}\right ) e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}{21 d} \]
command
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {15 \, \sqrt {2} \sqrt {-i} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} \sqrt {i} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (35 \, b^{3} \cos \left (d x + c\right )^{4} e^{\frac {3}{2}} + 135 \, a b^{2} \cos \left (d x + c\right )^{3} e^{\frac {3}{2}} + 7 \, {\left (27 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 15 \, {\left (7 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right ) e^{\frac {3}{2}} - 7 \, {\left (27 \, a^{2} b + 4 \, b^{3}\right )} e^{\frac {3}{2}}\right )} \sqrt {\sin \left (d x + c\right )}}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{3} e \cos \left (d x + c\right )^{3} + 3 \, a b^{2} e \cos \left (d x + c\right )^{2} + 3 \, a^{2} b e \cos \left (d x + c\right ) + a^{3} e\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ), x\right ) \]