\[ \int (c+d x) S(a+b x) \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{2} \pi }+\frac {d \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 b^{2} \pi }-\frac {d \FresnelC \left (b x +a \right )}{2 b^{2} \pi }-\frac {\left (-a d +b c \right )^{2} \mathrm {S}\left (b x +a \right )}{2 b^{2} d}+\frac {\left (d x +c \right )^{2} \mathrm {S}\left (b x +a \right )}{2 d} \]
command
integrate((d*x+c)*fresnel_sin(b*x+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\pi {\left (2 \, a b c - a^{2} d\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - \sqrt {b^{2}} d \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + {\left (\pi b^{3} d x^{2} + 2 \, \pi b^{3} c x\right )} \operatorname {S}\left (b x + a\right )}{2 \, \pi b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (d x + c\right )} {\rm fresnels}\left (b x + a\right ), x\right ) \]