\[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a +b \cos \left (x \right )+c \sin \left (x \right )\right )^{\frac {3}{2}} \left (c e \cos \left (x \right )-b e \sin \left (x \right )\right )}{5}-\frac {2 \left (c \left (3 a e +5 d \right ) \cos \left (x \right )-b \left (3 a e +5 d \right ) \sin \left (x \right )\right ) \sqrt {a +b \cos \left (x \right )+c \sin \left (x \right )}}{15}+\frac {2 \left (20 a d +3 a^{2} e +9 \left (b^{2}+c^{2}\right ) e \right ) \sqrt {\frac {\cos \left (x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {a +b \cos \left (x \right )+c \sin \left (x \right )}}{15 \cos \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) \sqrt {\frac {a +b \cos \left (x \right )+c \sin \left (x \right )}{a +\sqrt {b^{2}+c^{2}}}}}-\frac {2 \left (a^{2}-b^{2}-c^{2}\right ) \left (3 a e +5 d \right ) \sqrt {\frac {\cos \left (x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {\frac {a +b \cos \left (x \right )+c \sin \left (x \right )}{a +\sqrt {b^{2}+c^{2}}}}}{15 \cos \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) \sqrt {a +b \cos \left (x \right )+c \sin \left (x \right )}} \]
command
integrate((a+b*cos(x)+c*sin(x))^(3/2)*(d+b*e*cos(x)+c*e*sin(x)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left ({\left ({\left (b^{2} - c^{2}\right )} e \cos \left (x\right )^{2} + c^{2} e + a d + {\left (a b e + b d\right )} \cos \left (x\right ) + {\left (2 \, b c e \cos \left (x\right ) + a c e + c d\right )} \sin \left (x\right )\right )} \sqrt {b \cos \left (x\right ) + c \sin \left (x\right ) + a}, x\right ) \]