\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \, dx \]
Optimal antiderivative \[ \frac {128 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {904 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {128 a^{4} \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d}+\frac {150 a^{4} \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{77 d}+\frac {8 a^{4} \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9 d}+\frac {2 a^{4} \left (\cos ^{\frac {9}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{11 d}+\frac {904 a^{4} \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{231 d} \]
command
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (3390 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 3390 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 7392 i \, \sqrt {2} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 7392 i \, \sqrt {2} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (315 \, a^{4} \cos \left (d x + c\right )^{4} + 1540 \, a^{4} \cos \left (d x + c\right )^{3} + 3375 \, a^{4} \cos \left (d x + c\right )^{2} + 4928 \, a^{4} \cos \left (d x + c\right ) + 6780 \, a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a^{4} \cos \left (d x + c\right )^{5} + 4 \, a^{4} \cos \left (d x + c\right )^{4} + 6 \, a^{4} \cos \left (d x + c\right )^{3} + 4 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]