\[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{b \left (a^{2}-b^{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}}+\frac {2 \left (20 a^{2} b B -5 b^{3} B -3 a \,b^{2} \left (5 A -3 C \right )-24 a^{3} C \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 b^{3} \left (a^{2}-b^{2}\right ) d}+\frac {2 \left (5 A \,b^{2}-5 a b B +6 a^{2} C -b^{2} C \right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{5 b^{2} \left (a^{2}-b^{2}\right ) d}-\frac {2 \left (40 a^{3} b B -25 a \,b^{3} B -6 a^{2} b^{2} \left (5 A -4 C \right )-48 a^{4} C +3 b^{4} \left (5 A +3 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} \left (a^{2}-b^{2}\right ) d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}+\frac {2 \left (40 a^{2} b B +5 b^{3} B -48 a^{3} C -6 a \,b^{2} \left (5 A +2 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(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 \, C a^{4} b^{2} - 20 \, B a^{3} b^{3} + 3 \, {\left (5 \, A - 3 \, C\right )} a^{2} b^{4} + 5 \, B a b^{5} - 3 \, {\left (C a^{2} b^{4} - C b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, C a^{3} b^{3} - 5 \, B a^{2} b^{4} - 6 \, C a b^{5} + 5 \, B 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 \, C a^{5} b + 80 i \, B a^{4} b^{2} - 12 i \, {\left (5 \, A - 7 \, C\right )} a^{3} b^{3} - 80 i \, B a^{2} b^{4} + 3 i \, {\left (25 \, A + 9 \, C\right )} a b^{5} - 15 i \, B b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-96 i \, C a^{6} + 80 i \, B a^{5} b - 12 i \, {\left (5 \, A - 7 \, C\right )} a^{4} b^{2} - 80 i \, B a^{3} b^{3} + 3 i \, {\left (25 \, A + 9 \, C\right )} a^{2} b^{4} - 15 i \, B 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 \, C a^{5} b - 80 i \, B a^{4} b^{2} + 12 i \, {\left (5 \, A - 7 \, C\right )} a^{3} b^{3} + 80 i \, B a^{2} b^{4} - 3 i \, {\left (25 \, A + 9 \, C\right )} a b^{5} + 15 i \, B b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (96 i \, C a^{6} - 80 i \, B a^{5} b + 12 i \, {\left (5 \, A - 7 \, C\right )} a^{4} b^{2} + 80 i \, B a^{3} b^{3} - 3 i \, {\left (25 \, A + 9 \, C\right )} a^{2} b^{4} + 15 i \, B 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 \, C a^{4} b^{2} - 40 i \, B a^{3} b^{3} + 6 i \, {\left (5 \, A - 4 \, C\right )} a^{2} b^{4} + 25 i \, B a b^{5} - 3 i \, {\left (5 \, A + 3 \, C\right )} b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (48 i \, C a^{5} b - 40 i \, B a^{4} b^{2} + 6 i \, {\left (5 \, A - 4 \, C\right )} a^{3} b^{3} + 25 i \, B a^{2} b^{4} - 3 i \, {\left (5 \, A + 3 \, C\right )} 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 \, C a^{4} b^{2} + 40 i \, B a^{3} b^{3} - 6 i \, {\left (5 \, A - 4 \, C\right )} a^{2} b^{4} - 25 i \, B a b^{5} + 3 i \, {\left (5 \, A + 3 \, C\right )} b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-48 i \, C a^{5} b + 40 i \, B a^{4} b^{2} - 6 i \, {\left (5 \, A - 4 \, C\right )} a^{3} b^{3} - 25 i \, B a^{2} b^{4} + 3 i \, {\left (5 \, A + 3 \, C\right )} 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
\[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{4} + B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \]