\[ \int \cos (c+d x) (a+b \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 b d} \]
command
integrate(cos(d*x+c)*(a+b*sin(d*x+c))**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} a^{\frac {5}{2}} x \cos {\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {a^{\frac {5}{2}} \sin {\left (c + d x \right )}}{d} & \text {for}\: b = 0 \\x \left (a + b \sin {\left (c \right )}\right )^{\frac {5}{2}} \cos {\left (c \right )} & \text {for}\: d = 0 \\\frac {2 a^{3} \sqrt {a + b \sin {\left (c + d x \right )}}}{7 b d} + \frac {6 a^{2} \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{7 d} + \frac {6 a b \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}}{7 d} + \frac {2 b^{2} \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{3}{\left (c + d x \right )}}{7 d} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________