\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {2 \left (110 a^{2} b B -539 b^{3} B -40 a^{3} C -5 a \,b^{2} \left (99 A +67 C \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{3465 b^{2} d}+\frac {2 \left (99 A \,b^{2}-22 a b B +8 a^{2} C +81 b^{2} C \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{693 b^{2} d}+\frac {2 \left (11 b B -4 a C \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{99 b^{2} d}+\frac {2 C \cos \left (d x +c \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{11 b d}-\frac {2 \left (110 a^{3} b B -1254 a \,b^{3} B -40 a^{4} C -75 b^{4} \left (11 A +9 C \right )-15 a^{2} b^{2} \left (33 A +19 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{3465 b^{2} d}-\frac {2 \left (110 a^{4} b B -3069 a^{2} b^{3} B -1617 b^{5} B -40 a^{5} C -15 a^{3} b^{2} \left (33 A +17 C \right )-15 a \,b^{4} \left (319 A +247 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{3465 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}+\frac {2 \left (a^{2}-b^{2}\right ) \left (110 a^{3} b B -1254 a \,b^{3} B -40 a^{4} C -75 b^{4} \left (11 A +9 C \right )-15 a^{2} b^{2} \left (33 A +19 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{3465 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (80 i \, C a^{6} - 220 i \, B a^{5} b + 30 i \, {\left (33 \, A + 16 \, C\right )} a^{4} b^{2} + 1023 i \, B a^{3} b^{3} - 15 i \, {\left (253 \, A + 169 \, C\right )} a^{2} b^{4} - 5379 i \, B a b^{5} - 225 i \, {\left (11 \, A + 9 \, C\right )} b^{6}\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 ) + \sqrt {2} {\left (-80 i \, C a^{6} + 220 i \, B a^{5} b - 30 i \, {\left (33 \, A + 16 \, C\right )} a^{4} b^{2} - 1023 i \, B a^{3} b^{3} + 15 i \, {\left (253 \, A + 169 \, C\right )} a^{2} b^{4} + 5379 i \, B a b^{5} + 225 i \, {\left (11 \, A + 9 \, C\right )} b^{6}\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 (-40 i \, C a^{5} b + 110 i \, B a^{4} b^{2} - 15 i \, {\left (33 \, A + 17 \, C\right )} a^{3} b^{3} - 3069 i \, B a^{2} b^{4} - 15 i \, {\left (319 \, A + 247 \, C\right )} a b^{5} - 1617 i \, B b^{6}\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 (40 i \, C a^{5} b - 110 i \, B a^{4} b^{2} + 15 i \, {\left (33 \, A + 17 \, C\right )} a^{3} b^{3} + 3069 i \, B a^{2} b^{4} + 15 i \, {\left (319 \, A + 247 \, C\right )} a b^{5} + 1617 i \, B b^{6}\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 (315 \, C b^{6} \cos \left (d x + c\right )^{4} - 20 \, C a^{4} b^{2} + 55 \, B a^{3} b^{3} + 5 \, {\left (297 \, A + 205 \, C\right )} a^{2} b^{4} + 1793 \, B a b^{5} + 75 \, {\left (11 \, A + 9 \, C\right )} b^{6} + 35 \, {\left (23 \, C a b^{5} + 11 \, B b^{6}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (113 \, C a^{2} b^{4} + 209 \, B a b^{5} + 9 \, {\left (11 \, A + 9 \, C\right )} b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, C a^{3} b^{3} + 825 \, B a^{2} b^{4} + 5 \, {\left (297 \, A + 229 \, C\right )} a b^{5} + 539 \, 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 )}{10395 \, b^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{5} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} + A a^{2} \cos \left (d x + c\right ) + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]