\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {-5 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )}{15 e \sqrt {2+3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )}}-\frac {\sqrt {\frac {\cos \left (e x -\arctan \left (\frac {5}{3}\right )+d \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ), \frac {\sqrt {510-30 \sqrt {34}}}{15}\right ) \sqrt {2+\sqrt {34}}}{15 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ) e} \]
command
integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {5 i + 3} {\left (-\left (9 i + 15\right ) \, \sqrt {2} \cos \left (x e + d\right ) - \left (15 i + 25\right ) \, \sqrt {2} \sin \left (x e + d\right ) - \left (6 i + 10\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 ) + \sqrt {-5 i + 3} {\left (\left (9 i - 15\right ) \, \sqrt {2} \cos \left (x e + d\right ) + \left (15 i - 25\right ) \, \sqrt {2} \sin \left (x e + d\right ) + \left (6 i - 10\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 ) - 51 \, \sqrt {5 i + 3} {\left (-3 i \, \sqrt {2} \cos \left (x e + d\right ) - 5 i \, \sqrt {2} \sin \left (x e + d\right ) - 2 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 ) - 51 \, \sqrt {-5 i + 3} {\left (3 i \, \sqrt {2} \cos \left (x e + d\right ) + 5 i \, \sqrt {2} \sin \left (x e + d\right ) + 2 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 (5 \, \cos \left (x e + d\right ) - 3 \, \sin \left (x e + d\right )\right )} \sqrt {3 \, \cos \left (x e + d\right ) + 5 \, \sin \left (x e + d\right ) + 2}}{1530 \, {\left (3 \, \cos \left (x e + d\right ) e + 5 \, e \sin \left (x e + d\right ) + 2 \, e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}{16 \, \cos \left (e x + d\right )^{2} - 10 \, {\left (3 \, \cos \left (e x + d\right ) + 2\right )} \sin \left (e x + d\right ) - 12 \, \cos \left (e x + d\right ) - 29}, x\right ) \]