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