\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {-5 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )}{75 e \left (2+3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )\right )^{\frac {5}{2}}}+\frac {\frac {8 \cos \left (e x +d \right )}{675}-\frac {8 \sin \left (e x +d \right )}{1125}}{e \left (2+3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )\right )^{\frac {3}{2}}}-\frac {199 \left (5 \cos \left (e x +d \right )-3 \sin \left (e x +d \right )\right )}{101250 e \sqrt {2+3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )}}-\frac {8 \sqrt {\frac {\cos \left (e x -\arctan \left (\frac {5}{3}\right )+d \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ), \frac {\sqrt {510-30 \sqrt {34}}}{15}\right )}{3375 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ) e \sqrt {2+\sqrt {34}}}-\frac {199 \sqrt {\frac {\cos \left (e x -\arctan \left (\frac {5}{3}\right )+d \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ), \frac {\sqrt {510-30 \sqrt {34}}}{15}\right ) \sqrt {2+\sqrt {34}}}{101250 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ) e} \]
command
integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {559 \, \sqrt {5 i + 3} {\left (-\left (594 i + 990\right ) \, \sqrt {2} \cos \left (x e + d\right )^{3} - \left (288 i + 480\right ) \, \sqrt {2} \cos \left (x e + d\right )^{2} + 5 \, {\left (\left (6 i + 10\right ) \, \sqrt {2} \cos \left (x e + d\right )^{2} + \left (108 i + 180\right ) \, \sqrt {2} \cos \left (x e + d\right ) + \left (111 i + 185\right ) \, \sqrt {2}\right )} \sin \left (x e + d\right ) + \left (783 i + 1305\right ) \, \sqrt {2} \cos \left (x e + d\right ) + \left (474 i + 790\right ) \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) - i \, \sin \left (x e + d\right ) - \frac {10}{51} i + \frac {2}{17}\right ) + 559 \, \sqrt {-5 i + 3} {\left (\left (594 i - 990\right ) \, \sqrt {2} \cos \left (x e + d\right )^{3} + \left (288 i - 480\right ) \, \sqrt {2} \cos \left (x e + d\right )^{2} + 5 \, {\left (-\left (6 i - 10\right ) \, \sqrt {2} \cos \left (x e + d\right )^{2} - \left (108 i - 180\right ) \, \sqrt {2} \cos \left (x e + d\right ) - \left (111 i - 185\right ) \, \sqrt {2}\right )} \sin \left (x e + d\right ) - \left (783 i - 1305\right ) \, \sqrt {2} \cos \left (x e + d\right ) - \left (474 i - 790\right ) \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) + i \, \sin \left (x e + d\right ) + \frac {10}{51} i + \frac {2}{17}\right ) + 10149 \, \sqrt {5 i + 3} {\left (198 i \, \sqrt {2} \cos \left (x e + d\right )^{3} + 96 i \, \sqrt {2} \cos \left (x e + d\right )^{2} + 5 \, {\left (-2 i \, \sqrt {2} \cos \left (x e + d\right )^{2} - 36 i \, \sqrt {2} \cos \left (x e + d\right ) - 37 i \, \sqrt {2}\right )} \sin \left (x e + d\right ) - 261 i \, \sqrt {2} \cos \left (x e + d\right ) - 158 i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) - i \, \sin \left (x e + d\right ) - \frac {10}{51} i + \frac {2}{17}\right )\right ) + 10149 \, \sqrt {-5 i + 3} {\left (-198 i \, \sqrt {2} \cos \left (x e + d\right )^{3} - 96 i \, \sqrt {2} \cos \left (x e + d\right )^{2} + 5 \, {\left (2 i \, \sqrt {2} \cos \left (x e + d\right )^{2} + 36 i \, \sqrt {2} \cos \left (x e + d\right ) + 37 i \, \sqrt {2}\right )} \sin \left (x e + d\right ) + 261 i \, \sqrt {2} \cos \left (x e + d\right ) + 158 i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) + i \, \sin \left (x e + d\right ) + \frac {10}{51} i + \frac {2}{17}\right )\right ) + 102 \, {\left (1990 \, \cos \left (x e + d\right )^{3} + 16680 \, \cos \left (x e + d\right )^{2} + {\left (39402 \, \cos \left (x e + d\right )^{2} + 8896 \, \cos \left (x e + d\right ) - 19923\right )} \sin \left (x e + d\right ) + 15295 \, \cos \left (x e + d\right ) - 8340\right )} \sqrt {3 \, \cos \left (x e + d\right ) + 5 \, \sin \left (x e + d\right ) + 2}}{10327500 \, {\left (198 \, \cos \left (x e + d\right )^{3} e + 96 \, \cos \left (x e + d\right )^{2} e - 261 \, \cos \left (x e + d\right ) e - 5 \, {\left (2 \, \cos \left (x e + d\right )^{2} e + 36 \, \cos \left (x e + d\right ) e + 37 \, e\right )} \sin \left (x e + d\right ) - 158 \, e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}{644 \, \cos \left (e x + d\right )^{4} + 1584 \, \cos \left (e x + d\right )^{3} + 284 \, \cos \left (e x + d\right )^{2} + 20 \, {\left (48 \, \cos \left (e x + d\right )^{3} - 4 \, \cos \left (e x + d\right )^{2} - 111 \, \cos \left (e x + d\right ) - 58\right )} \sin \left (e x + d\right ) - 1896 \, \cos \left (e x + d\right ) - 1241}, x\right ) \]