\[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {78 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {e \cos \left (d x +c \right )}}-\frac {78 a^{4} \sqrt {e \cos \left (d x +c \right )}}{7 d e}-\frac {2 a \left (a +a \sin \left (d x +c \right )\right )^{3} \sqrt {e \cos \left (d x +c \right )}}{7 d e}-\frac {26 \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )^{2} \sqrt {e \cos \left (d x +c \right )}}{35 d e}-\frac {78 \left (a^{4}+a^{4} \sin \left (d x +c \right )\right ) \sqrt {e \cos \left (d x +c \right )}}{35 d e} \]
command
integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (-195 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (28 \, a^{4} \cos \left (d x + c\right )^{2} - 280 \, a^{4} + 5 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 17 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{35 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e \cos \left (d x + c\right )}, x\right ) \]