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