\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (57 a^{2}+20 b^{2}\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{105 d e}+\frac {22 a b \left (a +b \cos \left (d x +c \right )\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{35 d e}+\frac {2 b \left (a +b \cos \left (d x +c \right )\right )^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{7 d e}-\frac {2 a \left (5 a^{2}+6 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \sin \left (d x +c \right )}}{5 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {\sin \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {21 i \, \sqrt {2} \sqrt {-i} {\left (5 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} \sqrt {i} {\left (5 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (15 \, b^{3} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 63 \, a b^{2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + 5 \, {\left (21 \, a^{2} b + 4 \, b^{3}\right )} e^{\frac {1}{2}}\right )} \sin \left (d x + c\right )^{\frac {3}{2}}}{105 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sqrt {e \sin \left (d x + c\right )}, x\right ) \]