\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {4 a \left (33 A \,b^{2}+8 a^{2} C +34 b^{2} C \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{1155 b^{3} d}+\frac {2 \left (8 a^{2} C +3 b^{2} \left (11 A +9 C \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{231 b^{3} d}-\frac {4 a C \cos \left (d x +c \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{33 b^{2} d}+\frac {2 C \left (\cos ^{2}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{11 b d}-\frac {2 \left (16 a^{4} C +6 a^{2} b^{2} \left (11 A +8 C \right )-25 b^{4} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{1155 b^{3} d}-\frac {4 a \left (8 a^{4} C +3 a^{2} b^{2} \left (11 A +6 C \right )-b^{4} \left (451 A +348 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{1155 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}+\frac {2 \left (a^{2}-b^{2}\right ) \left (16 a^{4} C +6 a^{2} b^{2} \left (11 A +8 C \right )-25 b^{4} \left (11 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{1155 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+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 (-32 i \, C a^{6} - 12 i \, {\left (11 \, A + 5 \, C\right )} a^{4} b^{2} + i \, {\left (121 \, A + 123 \, C\right )} a^{2} b^{4} - 75 i \, {\left (11 \, A + 9 \, C\right )} b^{6}\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 (32 i \, C a^{6} + 12 i \, {\left (11 \, A + 5 \, C\right )} a^{4} b^{2} - i \, {\left (121 \, A + 123 \, C\right )} a^{2} b^{4} + 75 i \, {\left (11 \, A + 9 \, C\right )} b^{6}\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 ) - 6 \, \sqrt {2} {\left (8 i \, C a^{5} b + 3 i \, {\left (11 \, A + 6 \, C\right )} a^{3} b^{3} - i \, {\left (451 \, A + 348 \, C\right )} a b^{5}\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 \, \sqrt {2} {\left (-8 i \, C a^{5} b - 3 i \, {\left (11 \, A + 6 \, C\right )} a^{3} b^{3} + i \, {\left (451 \, A + 348 \, C\right )} a b^{5}\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 (105 \, C b^{6} \cos \left (d x + c\right )^{4} + 140 \, C a b^{5} \cos \left (d x + c\right )^{3} + 8 \, C a^{4} b^{2} + {\left (33 \, A + 19 \, C\right )} a^{2} b^{4} + 25 \, {\left (11 \, A + 9 \, C\right )} b^{6} + 5 \, {\left (C a^{2} b^{4} + 3 \, {\left (11 \, A + 9 \, C\right )} b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (3 \, C a^{3} b^{3} - {\left (132 \, A + 101 \, C\right )} a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, b^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{5} + C a \cos \left (d x + c\right )^{4} + A b \cos \left (d x + c\right )^{3} + A a \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]