\[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}+a^{2} C \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 b \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {4 \left (3 A \,b^{4}-a^{2} b^{2} \left (A -6 C \right )-4 a^{4} C \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 b^{2} \left (a^{2}-b^{2}\right )^{2} d \sqrt {a +b \cos \left (d x +c \right )}}-\frac {4 a \left (a^{2} b^{2} \left (10 A -49 C \right )-b^{4} \left (20 A -7 C \right )+32 a^{4} C \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 b^{4} \left (a^{2}-b^{2}\right )^{2} d}+\frac {2 \left (a^{2} b^{2} \left (15 A -71 C \right )-b^{4} \left (35 A -3 C \right )+48 a^{4} C \right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 b^{3} \left (a^{2}-b^{2}\right )^{2} d}+\frac {2 \left (4 a^{4} b^{2} \left (10 A -53 C \right )-5 a^{2} b^{4} \left (15 A -11 C \right )+128 a^{6} C +3 b^{6} \left (5 A +3 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5} \left (a^{2}-b^{2}\right )^{2} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 a \left (4 a^{2} b^{2} \left (10 A -29 C \right )+128 a^{4} C -b^{4} \left (45 A +17 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5} \left (a^{2}-b^{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^3*(A+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
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, x\right ) \]