\[ \int (a \cos (c+d x)+b \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b \cos \left (d x +c \right )-a \sin \left (d x +c \right )\right ) \left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{5 d}+\frac {6 \left (a^{2}+b^{2}\right ) \sqrt {\frac {\cos \left (c +d x -\arctan \left (a , b\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (a , b\right )}{2}\right ), \sqrt {2}\right ) \sqrt {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}}{5 \cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (a , b\right )}{2}\right ) d \sqrt {\frac {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}{\sqrt {a^{2}+b^{2}}}}} \]
command
integrate((a*cos(d*x+c)+b*sin(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \sqrt {2} {\left (-i \, a^{2} - i \, b^{2}\right )} \sqrt {a - i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (i \, a^{2} + i \, b^{2}\right )} \sqrt {a + i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (2 \, a b \cos \left (d x + c\right )^{2} - {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - a b\right )} \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}}{5 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}, x\right ) \]