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