\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {\frac {2 c \cos \left (e x +d \right )}{5}-\frac {2 b \sin \left (e x +d \right )}{5}}{\left (a^{2}-b^{2}-c^{2}\right ) e \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{\frac {5}{2}}}+\frac {\frac {16 a c \cos \left (e x +d \right )}{15}-\frac {16 a b \sin \left (e x +d \right )}{15}}{\left (a^{2}-b^{2}-c^{2}\right )^{2} e \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{\frac {3}{2}}}+\frac {\frac {2 c \left (23 a^{2}+9 b^{2}+9 c^{2}\right ) \cos \left (e x +d \right )}{15}-\frac {2 b \left (23 a^{2}+9 b^{2}+9 c^{2}\right ) \sin \left (e x +d \right )}{15}}{\left (a^{2}-b^{2}-c^{2}\right )^{3} e \sqrt {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}}+\frac {2 \left (23 a^{2}+9 b^{2}+9 c^{2}\right ) \sqrt {\frac {\cos \left (d +e x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}}{15 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) \left (a^{2}-b^{2}-c^{2}\right )^{3} e \sqrt {\frac {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}{a +\sqrt {b^{2}+c^{2}}}}}-\frac {16 a \sqrt {\frac {\cos \left (d +e x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {\frac {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}{a +\sqrt {b^{2}+c^{2}}}}}{15 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) \left (a^{2}-b^{2}-c^{2}\right )^{2} e \sqrt {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}} \]
command
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),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 \[ {\rm integral}\left (\frac {\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}{{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{4} + a^{4} + 6 \, a^{2} c^{2} + c^{4} + 4 \, {\left (a b^{3} - 3 \, a b c^{2}\right )} \cos \left (e x + d\right )^{3} + 2 \, {\left (3 \, a^{2} b^{2} - c^{4} - 3 \, {\left (a^{2} - b^{2}\right )} c^{2}\right )} \cos \left (e x + d\right )^{2} + 4 \, {\left (a^{3} b + 3 \, a b c^{2}\right )} \cos \left (e x + d\right ) + 4 \, {\left (a^{3} c + a c^{3} + {\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{3} + {\left (3 \, a b^{2} c - a c^{3}\right )} \cos \left (e x + d\right )^{2} + {\left (3 \, a^{2} b c + b c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}, x\right ) \]