\[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (9 a^{2}+2 b^{2}\right ) e \cos \left (d x +c \right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{45 d}+\frac {22 a b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{63 d e}+\frac {2 b \left (a +b \cos \left (d x +c \right )\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{9 d e}-\frac {2 \left (9 a^{2}+2 b^{2}\right ) e^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \sin \left (d x +c \right )}}{15 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {\sin \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {21 i \, \sqrt {2} \sqrt {-i} {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} \sqrt {i} {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (35 \, b^{2} \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} + 90 \, a b \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} - 90 \, a b e^{\frac {5}{2}} + 21 \, {\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right ) e^{\frac {5}{2}}\right )} \sin \left (d x + c\right )^{\frac {3}{2}}}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b^{2} e^{2} \cos \left (d x + c\right )^{4} + 2 \, a b e^{2} \cos \left (d x + c\right )^{3} - 2 \, a b e^{2} \cos \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sqrt {e \sin \left (d x + c\right )}, x\right ) \]