\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (8 a^{2} C +7 b^{2} \left (9 A +7 C \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{315 b^{2} d}-\frac {8 a C \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{63 b^{2} d}+\frac {2 C \cos \left (d x +c \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{9 b d}+\frac {2 a \left (63 A \,b^{2}+8 a^{2} C +39 b^{2} C \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 b^{2} d}+\frac {2 \left (8 a^{4} C +21 b^{4} \left (9 A +7 C \right )+3 a^{2} b^{2} \left (21 A +11 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 a \left (a^{2}-b^{2}\right ) \left (63 A \,b^{2}+8 a^{2} C +39 b^{2} C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)*(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 {2 \, \sqrt {2} {\left (-8 i \, C a^{5} - 3 i \, {\left (21 \, A + 10 \, C\right )} a^{3} b^{2} + 3 i \, {\left (63 \, A + 44 \, C\right )} a b^{4}\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 \, \sqrt {2} {\left (8 i \, C a^{5} + 3 i \, {\left (21 \, A + 10 \, C\right )} a^{3} b^{2} - 3 i \, {\left (63 \, A + 44 \, C\right )} a b^{4}\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 (-8 i \, C a^{4} b - 3 i \, {\left (21 \, A + 11 \, C\right )} a^{2} b^{3} - 21 i \, {\left (9 \, A + 7 \, C\right )} 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 ) + 3 \, \sqrt {2} {\left (8 i \, C a^{4} b + 3 i \, {\left (21 \, A + 11 \, C\right )} a^{2} b^{3} + 21 i \, {\left (9 \, A + 7 \, C\right )} 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 (35 \, C b^{5} \cos \left (d x + c\right )^{3} + 50 \, C a b^{4} \cos \left (d x + c\right )^{2} - 4 \, C a^{3} b^{2} + 2 \, {\left (63 \, A + 44 \, C\right )} a b^{4} + {\left (3 \, C a^{2} b^{3} + 7 \, {\left (9 \, A + 7 \, C\right )} 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 )}{945 \, b^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{4} + C a \cos \left (d x + c\right )^{3} + A b \cos \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]