\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx \]
Optimal antiderivative \[ -\frac {18 a b \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{35 d e}-\frac {2 b \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (a +b \sin \left (d x +c \right )\right )}{7 d e}+\frac {2 \left (7 a^{2}+2 b^{2}\right ) e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {e \cos \left (d x +c \right )}}+\frac {2 \left (7 a^{2}+2 b^{2}\right ) e \sin \left (d x +c \right ) \sqrt {e \cos \left (d x +c \right )}}{21 d} \]
command
integrate((e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} {\left (7 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} {\left (7 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (42 \, a b \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 5 \, {\left (3 \, b^{2} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} - {\left (7 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {3}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{105 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b^{2} e \cos \left (d x + c\right )^{3} - 2 \, a b e \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} e \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]