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