\[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 b \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {8 a^{2} \left (2 a^{2}-3 b^{2}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 b^{2} \left (a^{2}-b^{2}\right )^{2} d \sqrt {a +b \cos \left (d x +c \right )}}-\frac {4 a \left (32 a^{4}-49 a^{2} b^{2}+7 b^{4}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 b^{4} \left (a^{2}-b^{2}\right )^{2} d}+\frac {2 \left (48 a^{4}-71 a^{2} b^{2}+3 b^{4}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 b^{3} \left (a^{2}-b^{2}\right )^{2} d}+\frac {2 \left (128 a^{6}-212 a^{4} b^{2}+55 a^{2} b^{4}+9 b^{6}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5} \left (a^{2}-b^{2}\right )^{2} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 a \left (128 a^{4}-116 a^{2} b^{2}-17 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5} \left (a^{2}-b^{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {6 \, {\left (64 \, a^{7} b^{2} - 98 \, a^{5} b^{4} + 14 \, a^{3} b^{6} - 3 \, {\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (16 \, a^{6} b^{3} - 25 \, a^{4} b^{5} + 5 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + 2 \, {\left (\sqrt {2} {\left (-128 i \, a^{7} b^{2} + 260 i \, a^{5} b^{4} - 121 i \, a^{3} b^{6} - 21 i \, a b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-128 i \, a^{8} b + 260 i \, a^{6} b^{3} - 121 i \, a^{4} b^{5} - 21 i \, a^{2} b^{7}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-128 i \, a^{9} + 260 i \, a^{7} b^{2} - 121 i \, a^{5} b^{4} - 21 i \, a^{3} b^{6}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + 2 \, {\left (\sqrt {2} {\left (128 i \, a^{7} b^{2} - 260 i \, a^{5} b^{4} + 121 i \, a^{3} b^{6} + 21 i \, a b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (128 i \, a^{8} b - 260 i \, a^{6} b^{3} + 121 i \, a^{4} b^{5} + 21 i \, a^{2} b^{7}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (128 i \, a^{9} - 260 i \, a^{7} b^{2} + 121 i \, a^{5} b^{4} + 21 i \, a^{3} b^{6}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + 3 \, {\left (\sqrt {2} {\left (-128 i \, a^{6} b^{3} + 212 i \, a^{4} b^{5} - 55 i \, a^{2} b^{7} - 9 i \, b^{9}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-128 i \, a^{7} b^{2} + 212 i \, a^{5} b^{4} - 55 i \, a^{3} b^{6} - 9 i \, a b^{8}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-128 i \, a^{8} b + 212 i \, a^{6} b^{3} - 55 i \, a^{4} b^{5} - 9 i \, a^{2} b^{7}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, {\left (\sqrt {2} {\left (128 i \, a^{6} b^{3} - 212 i \, a^{4} b^{5} + 55 i \, a^{2} b^{7} + 9 i \, b^{9}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (128 i \, a^{7} b^{2} - 212 i \, a^{5} b^{4} + 55 i \, a^{3} b^{6} + 9 i \, a b^{8}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (128 i \, a^{8} b - 212 i \, a^{6} b^{3} + 55 i \, a^{4} b^{5} + 9 i \, a^{2} b^{7}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right )}{45 \, {\left ({\left (a^{4} b^{8} - 2 \, a^{2} b^{10} + b^{12}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{7} - 2 \, a^{3} b^{9} + a b^{11}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{6} - 2 \, a^{4} b^{8} + a^{2} b^{10}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{5}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, x\right ) \]