\[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {4 a \left (42 A \,b^{2}+32 a^{2} C +31 b^{2} C \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 b^{4} d}+\frac {2 \left (48 a^{2} C +7 b^{2} \left (9 A +7 C \right )\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 b^{3} d}-\frac {16 a C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{63 b^{2} d}+\frac {2 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{9 b d}+\frac {2 \left (128 a^{4} C +21 b^{4} \left (9 A +7 C \right )+12 a^{2} b^{2} \left (14 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 a \left (128 a^{4} C +4 a^{2} b^{2} \left (42 A +19 C \right )+3 b^{4} \left (49 A +37 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {2} {\left (-128 i \, C a^{5} - 12 i \, {\left (14 \, A + 5 \, C\right )} a^{3} b^{2} - 3 i \, {\left (42 \, A + 31 \, C\right )} a b^{4}\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 ) + 2 \, \sqrt {2} {\left (128 i \, C a^{5} + 12 i \, {\left (14 \, A + 5 \, C\right )} a^{3} b^{2} + 3 i \, {\left (42 \, A + 31 \, C\right )} a b^{4}\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 \, \sqrt {2} {\left (-128 i \, C a^{4} b - 12 i \, {\left (14 \, A + 9 \, C\right )} a^{2} b^{3} - 21 i \, {\left (9 \, A + 7 \, C\right )} b^{5}\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 \, \sqrt {2} {\left (128 i \, C a^{4} b + 12 i \, {\left (14 \, A + 9 \, C\right )} a^{2} b^{3} + 21 i \, {\left (9 \, A + 7 \, C\right )} b^{5}\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 ) - 6 \, {\left (35 \, C b^{5} \cos \left (d x + c\right )^{3} - 40 \, C a b^{4} \cos \left (d x + c\right )^{2} - 64 \, C a^{3} b^{2} - 2 \, {\left (42 \, A + 31 \, C\right )} a b^{4} + {\left (48 \, C a^{2} b^{3} + 7 \, {\left (9 \, A + 7 \, C\right )} 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 )}{945 \, b^{6} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}}, x\right ) \]