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