\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx \]
Optimal antiderivative \[ -\frac {4 a \,b^{3} \left (175 A -27 C \right ) \sin \left (d x +c \right )}{105 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 A \left (a +b \cos \left (d x +c \right )\right )^{4} \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}-\frac {2 b^{2} \left (3 a^{2} \left (49 A -13 C \right )-b^{2} \left (7 A +5 C \right )\right ) \sin \left (d x +c \right )}{21 d \sqrt {\sec \left (d x +c \right )}}-\frac {2 b^{2} \left (21 A -C \right ) \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{7 d \sqrt {\sec \left (d x +c \right )}}+\frac {16 A b \left (a +b \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{3 d}-\frac {8 a b \left (5 a^{2} \left (A -C \right )-b^{2} \left (5 A +3 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (42 a^{2} b^{2} \left (3 A +C \right )+7 a^{4} \left (A +3 C \right )+b^{4} \left (7 A +5 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \sqrt {2} {\left (7 i \, {\left (A + 3 \, C\right )} a^{4} + 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} + i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (A + 3 \, C\right )} a^{4} - 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} - i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 \, \sqrt {2} {\left (5 i \, {\left (A - C\right )} a^{3} b - i \, {\left (5 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 \, \sqrt {2} {\left (-5 i \, {\left (A - C\right )} a^{3} b + i \, {\left (5 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, C b^{4} \cos \left (d x + c\right )^{4} + 84 \, C a b^{3} \cos \left (d x + c\right )^{3} + 420 \, A a^{3} b \cos \left (d x + c\right ) + 35 \, A a^{4} + 5 \, {\left (42 \, C a^{2} b^{2} + {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sec \left (d x + c\right )^{\frac {5}{2}}, x\right ) \]