\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {130 \,13^{\frac {3}{4}} \sqrt {\frac {\cos \left (c +d x -\arctan \left (\frac {3}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ) d}-\frac {2 \left (3 \cos \left (d x +c \right )-2 \sin \left (d x +c \right )\right ) \left (2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{7 d}-\frac {130 \left (3 \cos \left (d x +c \right )-2 \sin \left (d x +c \right )\right ) \sqrt {2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )}}{21 d} \]
command
integrate((2*cos(d*x+c)+3*sin(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\left (130 i + 195\right ) \, \sqrt {3 i + 2} \sqrt {2} {\rm weierstrassPInverse}\left (\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - \left (130 i - 195\right ) \, \sqrt {2} \sqrt {-3 i + 2} {\rm weierstrassPInverse}\left (-\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (27 \, \cos \left (d x + c\right )^{3} + 46 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) + 204 \, \cos \left (d x + c\right )\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{21 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (46 \, \cos \left (d x + c\right )^{3} - 9 \, {\left (\cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) - 54 \, \cos \left (d x + c\right )\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}, x\right ) \]