\[ \int (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (45 a b B -10 a^{2} C +49 b^{2} C \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{315 b d}+\frac {2 \left (9 b B -2 a C \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{63 b d}+\frac {2 C \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{9 b d}+\frac {2 \left (45 a^{2} b B +75 b^{3} B -10 a^{3} C +114 C a \,b^{2}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 b d}+\frac {2 \left (45 a^{3} b B +435 a \,b^{3} B -10 a^{4} C +279 a^{2} b^{2} C +147 C \,b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 \left (a^{2}-b^{2}\right ) \left (45 a^{2} b B +75 b^{3} B -10 a^{3} C +114 C a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))^(5/2)*(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 (-20 i \, C a^{5} + 90 i \, B a^{4} b + 93 i \, C a^{3} b^{2} - 345 i \, B a^{2} b^{3} - 489 i \, C a b^{4} - 225 i \, B b^{5}\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 (20 i \, C a^{5} - 90 i \, B a^{4} b - 93 i \, C a^{3} b^{2} + 345 i \, B a^{2} b^{3} + 489 i \, C a b^{4} + 225 i \, B b^{5}\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 (10 i \, C a^{4} b - 45 i \, B a^{3} b^{2} - 279 i \, C a^{2} b^{3} - 435 i \, B a b^{4} - 147 i \, C 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 (-10 i \, C a^{4} b + 45 i \, B a^{3} b^{2} + 279 i \, C a^{2} b^{3} + 435 i \, B a b^{4} + 147 i \, C 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} + 5 \, C a^{3} b^{2} + 135 \, B a^{2} b^{3} + 163 \, C a b^{4} + 75 \, B b^{5} + 5 \, {\left (19 \, C a b^{4} + 9 \, B b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (75 \, C a^{2} b^{3} + 135 \, B a b^{4} + 49 \, 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 )}{945 \, b^{3} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{4} + B a^{2} \cos \left (d x + c\right ) + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]