\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \sin \left (d x +c \right )}{5 b \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {2 \left (3 a^{2} b B +5 b^{3} B +2 a^{3} C -2 a \,b^{2} \left (4 A +5 C \right )\right ) \sin \left (d x +c \right )}{15 b \left (a^{2}-b^{2}\right )^{2} d \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 \left (3 a^{3} b B +29 a \,b^{3} B +2 a^{4} C -3 b^{4} \left (3 A +5 C \right )-a^{2} b^{2} \left (23 A +19 C \right )\right ) \sin \left (d x +c \right )}{15 b \left (a^{2}-b^{2}\right )^{3} d \sqrt {a +b \cos \left (d x +c \right )}}-\frac {2 \left (3 a^{3} b B +29 a \,b^{3} B +2 a^{4} C -3 b^{4} \left (3 A +5 C \right )-a^{2} b^{2} \left (23 A +19 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} \left (a^{2}-b^{2}\right )^{3} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}+\frac {2 \left (3 a^{2} b B +5 b^{3} B +2 a^{3} C -2 a \,b^{2} \left (4 A +5 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} \left (a^{2}-b^{2}\right )^{2} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{4} \cos \left (d x + c\right )^{4} + 4 \, a b^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 4 \, a^{3} b \cos \left (d x + c\right ) + a^{4}}, x\right ) \]