\[ \int \frac {1}{(2 \cos (c+d x)+3 \sin (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \,13^{\frac {3}{4}} \sqrt {\frac {\cos \left (c +d x -\arctan \left (\frac {3}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ), \sqrt {2}\right )}{507 \cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ) d}-\frac {2 \left (3 \cos \left (d x +c \right )-2 \sin \left (d x +c \right )\right )}{39 d \left (2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate(1/(2*cos(d*x+c)+3*sin(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {\sqrt {3 i + 2} {\left (-\left (10 i + 15\right ) \, \sqrt {2} \cos \left (d x + c\right )^{2} + \left (24 i + 36\right ) \, \sqrt {2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \left (18 i + 27\right ) \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + \sqrt {-3 i + 2} {\left (\left (10 i - 15\right ) \, \sqrt {2} \cos \left (d x + c\right )^{2} - \left (24 i - 36\right ) \, \sqrt {2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \left (18 i - 27\right ) \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 26 \, {\left (3 \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right )\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{507 \, {\left (5 \, d \cos \left (d x + c\right )^{2} - 12 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 9 \, d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{46 \, \cos \left (d x + c\right )^{3} - 9 \, {\left (\cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) - 54 \, \cos \left (d x + c\right )}, x\right ) \]