Optimal antiderivative \[ \frac {x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b \pi }+\frac {3 \,\mathrm {S}\left (b x \right )}{4 b^{4} \pi ^{2}}+\frac {x^{4} \mathrm {S}\left (b x \right )}{4}-\frac {3 x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{3} \pi ^{2}} \]
command
integrate(x^3*fresnel_sin(b*x),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{4} \, x^{4} \operatorname {S}\left (b x\right ) + \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 12 \, \sqrt {\frac {1}{2}} \pi b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (3 i + 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (3 i - 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{8 \, \pi ^{3} b^{4}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int x^{3} {\rm fresnels}\left (b x\right )\,{d x} \]________________________________________________________________________________________