\[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (5 \cos \left (e x +d \right )-3 \sin \left (e x +d \right )\right ) \left (2+3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )\right )^{\frac {3}{2}}}{5 e}-\frac {32 \left (5 \cos \left (e x +d \right )-3 \sin \left (e x +d \right )\right ) \sqrt {2+3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )}}{15 e}+\frac {64 \sqrt {\frac {\cos \left (e x -\arctan \left (\frac {5}{3}\right )+d \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ), \frac {\sqrt {510-30 \sqrt {34}}}{15}\right )}{\cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ) e \sqrt {2+\sqrt {34}}}+\frac {796 \sqrt {\frac {\cos \left (e x -\arctan \left (\frac {5}{3}\right )+d \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ), \frac {\sqrt {510-30 \sqrt {34}}}{15}\right ) \sqrt {2+\sqrt {34}}}{15 \cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (\frac {5}{3}\right )}{2}\right ) e} \]
command
integrate((2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2}{765} \, {\left (-\left (1677 i + 2795\right ) \, \sqrt {5 i + 3} \sqrt {2} {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) - i \, \sin \left (x e + d\right ) - \frac {10}{51} i + \frac {2}{17}\right ) + \left (1677 i - 2795\right ) \, \sqrt {-5 i + 3} \sqrt {2} {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) + i \, \sin \left (x e + d\right ) + \frac {10}{51} i + \frac {2}{17}\right ) + 10149 i \, \sqrt {5 i + 3} \sqrt {2} {\rm weierstrassZeta}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) - i \, \sin \left (x e + d\right ) - \frac {10}{51} i + \frac {2}{17}\right )\right ) - 10149 i \, \sqrt {-5 i + 3} \sqrt {2} {\rm weierstrassZeta}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (x e + d\right ) + i \, \sin \left (x e + d\right ) + \frac {10}{51} i + \frac {2}{17}\right )\right ) + 51 \, {\left (90 \, \cos \left (x e + d\right )^{2} + 6 \, {\left (8 \, \cos \left (x e + d\right ) - 11\right )} \sin \left (x e + d\right ) + 110 \, \cos \left (x e + d\right ) - 45\right )} \sqrt {3 \, \cos \left (x e + d\right ) + 5 \, \sin \left (x e + d\right ) + 2}\right )} e^{\left (-1\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (16 \, \cos \left (e x + d\right )^{2} - 10 \, {\left (3 \, \cos \left (e x + d\right ) + 2\right )} \sin \left (e x + d\right ) - 12 \, \cos \left (e x + d\right ) - 29\right )} \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}, x\right ) \]