\[ \int \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {4 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 a^{2} d}-\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {11}{2}}}{11 a^{3} d} \]
command
integrate(cos(d*x+c)**3*(a+a*sin(d*x+c))**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {8 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{5}{\left (c + d x \right )}}{77 d} + \frac {272 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{4}{\left (c + d x \right )}}{693 d} + \frac {2 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {368 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{3}{\left (c + d x \right )}}{693 d} + \frac {6 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {64 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{231 d} + \frac {6 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {8 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{693 d} + \frac {2 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}}{7 d} - \frac {16 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a}}{693 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{\frac {5}{2}} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________