\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (520 a^{3} b B +4355 a \,b^{3} B -240 a^{4} C +539 b^{4} \left (13 A +11 C \right )-10 a^{2} b^{2} \left (143 A +124 C \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{45045 b^{3} d}+\frac {2 \left (104 a^{2} b B +1053 b^{3} B -48 a^{3} C -2 a \,b^{2} \left (143 A +166 C \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{9009 b^{3} d}+\frac {2 \left (143 A \,b^{2}-52 a b B +24 a^{2} C +121 b^{2} C \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{1287 b^{3} d}+\frac {2 \left (13 b B -6 a C \right ) \cos \left (d x +c \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{143 b^{2} d}+\frac {2 C \left (\cos ^{2}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{13 b d}+\frac {2 \left (520 a^{4} b B +3705 a^{2} b^{3} B +8775 b^{5} B -240 a^{5} C -10 a^{3} b^{2} \left (143 A +94 C \right )+6 a \,b^{4} \left (2717 A +2174 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{45045 b^{3} d}+\frac {2 \left (520 a^{5} b B +3315 a^{3} b^{3} B +48165 a \,b^{5} B -240 a^{6} C +1617 b^{6} \left (13 A +11 C \right )-10 a^{4} b^{2} \left (143 A +76 C \right )+3 a^{2} b^{4} \left (13299 A +10223 C \right )\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 )}}{45045 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 \left (a^{2}-b^{2}\right ) \left (520 a^{4} b B +3705 a^{2} b^{3} B +8775 b^{5} B -240 a^{5} C -10 a^{3} b^{2} \left (143 A +94 C \right )+6 a \,b^{4} \left (2717 A +2174 C \right )\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}}}{45045 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (-480 i \, C a^{7} + 1040 i \, B a^{6} b - 20 i \, {\left (143 \, A + 67 \, C\right )} a^{5} b^{2} + 6240 i \, B a^{4} b^{3} + 3 i \, {\left (4433 \, A + 3761 \, C\right )} a^{3} b^{4} - 32955 i \, B a^{2} b^{5} - 3 i \, {\left (23309 \, A + 18973 \, C\right )} a b^{6} - 26325 i \, B b^{7}\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 ) + \sqrt {2} {\left (480 i \, C a^{7} - 1040 i \, B a^{6} b + 20 i \, {\left (143 \, A + 67 \, C\right )} a^{5} b^{2} - 6240 i \, B a^{4} b^{3} - 3 i \, {\left (4433 \, A + 3761 \, C\right )} a^{3} b^{4} + 32955 i \, B a^{2} b^{5} + 3 i \, {\left (23309 \, A + 18973 \, C\right )} a b^{6} + 26325 i \, B b^{7}\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 \, \sqrt {2} {\left (240 i \, C a^{6} b - 520 i \, B a^{5} b^{2} + 10 i \, {\left (143 \, A + 76 \, C\right )} a^{4} b^{3} - 3315 i \, B a^{3} b^{4} - 3 i \, {\left (13299 \, A + 10223 \, C\right )} a^{2} b^{5} - 48165 i \, B a b^{6} - 1617 i \, {\left (13 \, A + 11 \, C\right )} b^{7}\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 \, \sqrt {2} {\left (-240 i \, C a^{6} b + 520 i \, B a^{5} b^{2} - 10 i \, {\left (143 \, A + 76 \, C\right )} a^{4} b^{3} + 3315 i \, B a^{3} b^{4} + 3 i \, {\left (13299 \, A + 10223 \, C\right )} a^{2} b^{5} + 48165 i \, B a b^{6} + 1617 i \, {\left (13 \, A + 11 \, C\right )} b^{7}\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 ) + 6 \, {\left (3465 \, C b^{7} \cos \left (d x + c\right )^{5} + 120 \, C a^{5} b^{2} - 260 \, B a^{4} b^{3} + 5 \, {\left (143 \, A + 79 \, C\right )} a^{3} b^{4} + 13325 \, B a^{2} b^{5} + {\left (23309 \, A + 18973 \, C\right )} a b^{6} + 8775 \, B b^{7} + 315 \, {\left (27 \, C a b^{6} + 13 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + 35 \, {\left (159 \, C a^{2} b^{5} + 299 \, B a b^{6} + 11 \, {\left (13 \, A + 11 \, C\right )} b^{7}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (15 \, C a^{3} b^{4} + 1469 \, B a^{2} b^{5} + {\left (2717 \, A + 2209 \, C\right )} a b^{6} + 1053 \, B b^{7}\right )} \cos \left (d x + c\right )^{2} - {\left (90 \, C a^{4} b^{3} - 195 \, B a^{3} b^{4} - 15 \, {\left (715 \, A + 543 \, C\right )} a^{2} b^{5} - 14885 \, B a b^{6} - 539 \, {\left (13 \, A + 11 \, C\right )} 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 )}{135135 \, b^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{6} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{5} + A a^{2} \cos \left (d x + c\right )^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]