\[ \int (c+d x)^2 S(a+b x) \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{3} \pi }+\frac {d \left (-a d +b c \right ) \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{3} \pi }+\frac {d^{2} \left (b x +a \right )^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 b^{3} \pi }-\frac {d \left (-a d +b c \right ) \FresnelC \left (b x +a \right )}{b^{3} \pi }-\frac {\left (-a d +b c \right )^{3} \mathrm {S}\left (b x +a \right )}{3 b^{3} d}+\frac {\left (d x +c \right )^{3} \mathrm {S}\left (b x +a \right )}{3 d}-\frac {2 d^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 b^{3} \pi ^{2}} \]
command
integrate((d*x+c)^2*fresnel_sin(b*x+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\pi ^{2} {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, b d^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - 3 \, \pi {\left (b c d - a d^{2}\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (\pi b^{3} d^{2} x^{2} + \pi {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x + \pi {\left (3 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + {\left (\pi ^{2} b^{4} d^{2} x^{3} + 3 \, \pi ^{2} b^{4} c d x^{2} + 3 \, \pi ^{2} b^{4} c^{2} x\right )} \operatorname {S}\left (b x + a\right )}{3 \, \pi ^{2} b^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} {\rm fresnels}\left (b x + a\right ), x\right ) \]