\[ \int \frac {1}{(5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {-3 \cos \! \left (e x +d \right )+4 \sin \! \left (e x +d \right )}{20 e \left (5+4 \cos \! \left (e x +d \right )+3 \sin \! \left (e x +d \right )\right )^{\frac {5}{2}}}-\frac {3 \left (3 \cos \! \left (e x +d \right )-4 \sin \! \left (e x +d \right )\right )}{400 e \left (5+4 \cos \! \left (e x +d \right )+3 \sin \! \left (e x +d \right )\right )^{\frac {3}{2}}}+\frac {3 \arctanh \! \left (\frac {\sin \left (d +e x -\arctan \left (\frac {3}{4}\right )\right ) \sqrt {2}}{2 \sqrt {1+\cos \left (d +e x -\arctan \left (\frac {3}{4}\right )\right )}}\right ) \sqrt {10}}{4000 e} \]
command
integrate(1/(5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {1}{4000} \, {\left (\frac {3 \, \sqrt {10} \log \left (\frac {{\left | -2 \, \sqrt {10} + 2 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 2 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \right |}}{{\left | 2 \, \sqrt {10} + 2 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 2 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \right |}}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3\right )} - \frac {20 \, {\left (797 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{7} - 7137 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{6} + 27543 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{5} - 30015 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{4} - 27105 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{3} - 7491 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} - 859 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 859 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 69\right )}}{{\left ({\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} - 6 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 6 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 1\right )}^{4} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3\right )}\right )} e^{\left (-1\right )} \]