Optimal antiderivative \[ -\frac {a^{3} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{4} \pi }+\frac {3 a^{2} \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 b^{4} \pi }-\frac {a \left (b x +a \right )^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{4} \pi }+\frac {\left (b x +a \right )^{3} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{4 b^{4} \pi }-\frac {3 a^{2} \FresnelC \left (b x +a \right )}{2 b^{4} \pi }-\frac {a^{4} \mathrm {S}\left (b x +a \right )}{4 b^{4}}+\frac {3 \,\mathrm {S}\left (b x +a \right )}{4 b^{4} \pi ^{2}}+\frac {x^{4} \mathrm {S}\left (b x +a \right )}{4}+\frac {2 a \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{b^{4} \pi ^{2}}-\frac {3 \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{4 b^{4} \pi ^{2}} \]
command
integrate(x^3*fresnel_sin(b*x+a),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{4} \, x^{4} \operatorname {S}\left (b x + a\right ) - \frac {{\left (16 \, {\left (\pi ^{2} e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + \pi ^{2} e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{4} + 32 \, {\left (-i \, \pi \Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + i \, \pi \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a^{2} + 16 \, {\left ({\left (\pi ^{2} e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + \pi ^{2} e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{3} + 2 \, {\left (-i \, \pi \Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + i \, \pi \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a\right )} b x - {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {5}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {5}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{4} - 12 \, {\left (\left (i - 1\right ) \, \sqrt {2} \pi \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \pi \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a^{2} - \left (4 i + 4\right ) \, \sqrt {2} \Gamma \left (\frac {5}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \left (4 i - 4\right ) \, \sqrt {2} \Gamma \left (\frac {5}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}}\right )} b}{32 \, {\left (\pi ^{3} b^{6} x + \pi ^{3} a b^{5}\right )}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int x^{3} {\rm fresnels}\left (b x + a\right )\,{d x} \]________________________________________________________________________________________