\[ \int \frac {(a+b \cos (c+d x))^3}{\sqrt {e \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 a \left (a^{2}+2 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\sin }\left (d x +c \right )\right )}{\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {e \sin \left (d x +c \right )}}+\frac {2 b \left (11 a^{2}+4 b^{2}\right ) \sqrt {e \sin \left (d x +c \right )}}{5 d e}+\frac {6 a b \left (a +b \cos \left (d x +c \right )\right ) \sqrt {e \sin \left (d x +c \right )}}{5 d e}+\frac {2 b \left (a +b \cos \left (d x +c \right )\right )^{2} \sqrt {e \sin \left (d x +c \right )}}{5 d e} \]
command
integrate((a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (5 \, \sqrt {2} \sqrt {-i} {\left (a^{3} + 2 \, a b^{2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} \sqrt {i} {\left (a^{3} + 2 \, a b^{2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (b^{3} \cos \left (d x + c\right )^{2} + 5 \, a b^{2} \cos \left (d x + c\right ) + 15 \, a^{2} b + 4 \, b^{3}\right )} \sqrt {\sin \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{5 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sqrt {e \sin \left (d x + c\right )}}{e \sin \left (d x + c\right )}, x\right ) \]