\[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \cos \left (e x +d \right )-b \sin \left (e x +d \right )\right ) \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{\frac {3}{2}}}{5 e}-\frac {16 \left (a c \cos \left (e x +d \right )-a b \sin \left (e x +d \right )\right ) \sqrt {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}}{15 e}+\frac {2 \left (23 a^{2}+9 b^{2}+9 c^{2}\right ) \sqrt {\frac {\cos \left (d +e x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}}{15 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) e \sqrt {\frac {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}{a +\sqrt {b^{2}+c^{2}}}}}-\frac {16 a \left (a^{2}-b^{2}-c^{2}\right ) \sqrt {\frac {\cos \left (d +e x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {\frac {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}{a +\sqrt {b^{2}+c^{2}}}}}{15 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) e \sqrt {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}} \]
command
integrate((a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),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 \[ {\rm integral}\left ({\left (2 \, a b \cos \left (e x + d\right ) + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} + a^{2} + c^{2} + 2 \, {\left (b c \cos \left (e x + d\right ) + a c\right )} \sin \left (e x + d\right )\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}, x\right ) \]