\[ \int (c+d x)^3 \text {FresnelC}(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {2 d^{2} \left (-a d +b c \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{4} \pi ^{2}}-\frac {3 d^{3} \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{4 b^{4} \pi ^{2}}-\frac {\left (-a d +b c \right )^{4} \FresnelC \left (b x +a \right )}{4 b^{4} d}+\frac {3 d^{3} \FresnelC \left (b x +a \right )}{4 b^{4} \pi ^{2}}+\frac {\left (d x +c \right )^{4} \FresnelC \left (b x +a \right )}{4 d}+\frac {3 d \left (-a d +b c \right )^{2} \mathrm {S}\left (b x +a \right )}{2 b^{4} \pi }-\frac {\left (-a d +b c \right )^{3} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{4} \pi }-\frac {3 d \left (-a d +b c \right )^{2} \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 b^{4} \pi }-\frac {d^{2} \left (-a d +b c \right ) \left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{4} \pi }-\frac {d^{3} \left (b x +a \right )^{3} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{4 b^{4} \pi } \]
command
integrate((d*x+c)^3*fresnel_cos(b*x+a),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {6 \, \pi {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (\pi ^{2} {\left (4 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 4 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} + 3 \, d^{3}\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (3 \, b^{2} d^{3} x + 8 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + {\left (\pi ^{2} b^{5} d^{3} x^{4} + 4 \, \pi ^{2} b^{5} c d^{2} x^{3} + 6 \, \pi ^{2} b^{5} c^{2} d x^{2} + 4 \, \pi ^{2} b^{5} c^{3} x\right )} \operatorname {C}\left (b x + a\right ) - {\left (\pi b^{4} d^{3} x^{3} + \pi {\left (4 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + \pi {\left (6 \, b^{4} c^{2} d - 4 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x + \pi {\left (4 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 4 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{4 \, \pi ^{2} b^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} {\rm fresnelc}\left (b x + a\right ), x\right ) \]