\[ \int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx \]
Optimal antiderivative \[ -\frac {\left (d x +c \right )^{\frac {5}{2}} \cos \left (2 b x +2 a \right )}{4 b}+\frac {5 d \left (d x +c \right )^{\frac {3}{2}} \sin \left (2 b x +2 a \right )}{16 b^{2}}-\frac {15 d^{\frac {5}{2}} \cos \left (2 a -\frac {2 b c}{d}\right ) \FresnelC \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}}{\sqrt {d}\, \sqrt {\pi }}\right ) \sqrt {\pi }}{128 b^{\frac {7}{2}}}+\frac {15 d^{\frac {5}{2}} \mathrm {S}\left (\frac {2 \sqrt {b}\, \sqrt {d x +c}}{\sqrt {d}\, \sqrt {\pi }}\right ) \sin \left (2 a -\frac {2 b c}{d}\right ) \sqrt {\pi }}{128 b^{\frac {7}{2}}}+\frac {15 d^{2} \cos \left (2 b x +2 a \right ) \sqrt {d x +c}}{64 b^{3}} \]
command
integrate((d*x+c)**(5/2)*cos(b*x+a)*sin(b*x+a),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {3 b^{\frac {3}{2}} \sqrt {\frac {d}{b}} \left (c + d x\right )^{\frac {9}{2}} \cos {\left (2 a - \frac {2 b c}{d} \right )} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {9}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {13}{4} \end {matrix}\middle | {- \frac {b^{2} \left (c + d x\right )^{2}}{d^{2}}} \right )}}{4 d^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {13}{4}\right )} - \frac {3 \sqrt {b} \sqrt {\frac {d}{b}} \left (c + d x\right )^{\frac {7}{2}} \sin {\left (2 a - \frac {2 b c}{d} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {11}{4} \end {matrix}\middle | {- \frac {b^{2} \left (c + d x\right )^{2}}{d^{2}}} \right )}}{8 d^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {\pi } \sqrt {\frac {d}{b}} \left (c + d x\right )^{3} \sin {\left (2 a - \frac {2 b c}{d} \right )} C\left (\frac {2 b \sqrt {c + d x}}{\sqrt {\pi } d \sqrt {\frac {b}{d}}}\right )}{2 d} + \frac {\sqrt {\pi } \sqrt {\frac {d}{b}} \left (c + d x\right )^{3} \cos {\left (2 a - \frac {2 b c}{d} \right )} S\left (\frac {2 b \sqrt {c + d x}}{\sqrt {\pi } d \sqrt {\frac {b}{d}}}\right )}{2 d} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________