\[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 a \left (A \,b^{2}+a^{2} C \right ) \sin \left (d x +c \right )}{b^{2} \left (a^{2}-b^{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}}+\frac {2 C \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{3 b^{2} d}-\frac {2 a \left (3 A \,b^{2}+8 a^{2} C -5 b^{2} C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} \left (a^{2}-b^{2}\right ) d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}+\frac {2 \left (3 A \,b^{2}+\left (8 a^{2}+b^{2}\right ) C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)*(A+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 (4 \, C a^{3} b^{2} + {\left (3 \, A - C\right )} a b^{4} + {\left (C a^{2} b^{3} - C b^{5}\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 (16 i \, C a^{4} b + 2 i \, {\left (3 \, A - 8 \, C\right )} a^{2} b^{3} - 3 i \, {\left (3 \, A + C\right )} b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (16 i \, C a^{5} + 2 i \, {\left (3 \, A - 8 \, C\right )} a^{3} b^{2} - 3 i \, {\left (3 \, A + C\right )} a 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 (-16 i \, C a^{4} b - 2 i \, {\left (3 \, A - 8 \, C\right )} a^{2} b^{3} + 3 i \, {\left (3 \, A + C\right )} b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-16 i \, C a^{5} - 2 i \, {\left (3 \, A - 8 \, C\right )} a^{3} b^{2} + 3 i \, {\left (3 \, A + C\right )} a 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 (-8 i \, C a^{3} b^{2} - i \, {\left (3 \, A - 5 \, C\right )} a b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-8 i \, C a^{4} b - i \, {\left (3 \, A - 5 \, C\right )} a^{2} b^{3}\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 (8 i \, C a^{3} b^{2} + i \, {\left (3 \, A - 5 \, C\right )} a b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (8 i \, C a^{4} b + i \, {\left (3 \, A - 5 \, C\right )} a^{2} b^{3}\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^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{4} - a b^{6}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \]