\[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 a \left (A b -B a \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{b \left (a^{2}-b^{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}}+\frac {2 \left (20 A \,a^{2} b -5 A \,b^{3}-24 a^{3} B +9 B a \,b^{2}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 b^{3} \left (a^{2}-b^{2}\right ) d}-\frac {2 \left (5 A a b -6 B \,a^{2}+b^{2} B \right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{5 b^{2} \left (a^{2}-b^{2}\right ) d}-\frac {2 \left (40 A \,a^{3} b -25 A a \,b^{3}-48 a^{4} B +24 B \,a^{2} b^{2}+9 b^{4} B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} \left (a^{2}-b^{2}\right ) d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}+\frac {2 \left (40 A \,a^{2} b +5 A \,b^{3}-48 a^{3} B -12 B a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {6 \, {\left (24 \, B a^{4} b^{2} - 20 \, A a^{3} b^{3} - 9 \, B a^{2} b^{4} + 5 \, A a b^{5} - 3 \, {\left (B a^{2} b^{4} - B b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4} - 6 \, B a b^{5} + 5 \, A b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + {\left (\sqrt {2} {\left (-96 i \, B a^{5} b + 80 i \, A a^{4} b^{2} + 84 i \, B a^{3} b^{3} - 80 i \, A a^{2} b^{4} + 27 i \, B a b^{5} - 15 i \, A b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-96 i \, B a^{6} + 80 i \, A a^{5} b + 84 i \, B a^{4} b^{2} - 80 i \, A a^{3} b^{3} + 27 i \, B a^{2} b^{4} - 15 i \, A a b^{5}\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 ) + {\left (\sqrt {2} {\left (96 i \, B a^{5} b - 80 i \, A a^{4} b^{2} - 84 i \, B a^{3} b^{3} + 80 i \, A a^{2} b^{4} - 27 i \, B a b^{5} + 15 i \, A b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (96 i \, B a^{6} - 80 i \, A a^{5} b - 84 i \, B a^{4} b^{2} + 80 i \, A a^{3} b^{3} - 27 i \, B a^{2} b^{4} + 15 i \, A a b^{5}\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 (48 i \, B a^{4} b^{2} - 40 i \, A a^{3} b^{3} - 24 i \, B a^{2} b^{4} + 25 i \, A a b^{5} - 9 i \, B b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (48 i \, B a^{5} b - 40 i \, A a^{4} b^{2} - 24 i \, B a^{3} b^{3} + 25 i \, A a^{2} b^{4} - 9 i \, B a b^{5}\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 (-48 i \, B a^{4} b^{2} + 40 i \, A a^{3} b^{3} + 24 i \, B a^{2} b^{4} - 25 i \, A a b^{5} + 9 i \, B b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-48 i \, B a^{5} b + 40 i \, A a^{4} b^{2} + 24 i \, B a^{3} b^{3} - 25 i \, A a^{2} b^{4} + 9 i \, B a b^{5}\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^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B \cos \left (d x + c\right )^{4} + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \]