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