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