\[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (\csc ^{\frac {5}{2}}\left (e x +d \right )\right ) \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right ) \left (a \cos \left (e x +d \right )-c \sin \left (e x +d \right )\right )}{3 \left (a^{2}-b^{2}+c^{2}\right ) e \left (a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )\right )^{\frac {5}{2}}}+\frac {8 \left (\csc ^{\frac {5}{2}}\left (e x +d \right )\right ) \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right )^{2} \left (a b \cos \left (e x +d \right )-b c \sin \left (e x +d \right )\right )}{3 \left (a^{2}-b^{2}+c^{2}\right )^{2} e \left (a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )\right )^{\frac {5}{2}}}+\frac {8 b \left (\csc ^{\frac {5}{2}}\left (e x +d \right )\right ) \sqrt {\frac {\cos \left (d +e x -\arctan \left (c , a\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {a^{2}+c^{2}}}{b +\sqrt {a^{2}+c^{2}}}}\right ) \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right )^{3}}{3 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ) \left (a^{2}-b^{2}+c^{2}\right )^{2} e \left (a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )\right )^{\frac {5}{2}} \sqrt {\frac {b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )}{b +\sqrt {a^{2}+c^{2}}}}}+\frac {2 \left (\csc ^{\frac {5}{2}}\left (e x +d \right )\right ) \sqrt {\frac {\cos \left (d +e x -\arctan \left (c , a\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {a^{2}+c^{2}}}{b +\sqrt {a^{2}+c^{2}}}}\right ) \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right )^{2} \sqrt {\frac {b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )}{b +\sqrt {a^{2}+c^{2}}}}}{3 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ) \left (a^{2}-b^{2}+c^{2}\right ) e \left (a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )\right )^{\frac {5}{2}}} \]
command
integrate(csc(e*x+d)^(5/2)/(a+c*cot(e*x+d)+b*csc(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 \[ {\rm integral}\left (\frac {\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \csc \left (e x + d\right )^{\frac {5}{2}}}{c^{3} \cot \left (e x + d\right )^{3} + b^{3} \csc \left (e x + d\right )^{3} + 3 \, a c^{2} \cot \left (e x + d\right )^{2} + 3 \, a^{2} c \cot \left (e x + d\right ) + a^{3} + 3 \, {\left (b^{2} c \cot \left (e x + d\right ) + a b^{2}\right )} \csc \left (e x + d\right )^{2} + 3 \, {\left (b c^{2} \cot \left (e x + d\right )^{2} + 2 \, a b c \cot \left (e x + d\right ) + a^{2} b\right )} \csc \left (e x + d\right )}, x\right ) \]