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="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\pi ^{2} b^{5} x^{4} \operatorname {S}\left (b x + a\right ) - 6 \, \pi a^{2} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (\pi ^{2} a^{4} - 3\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (\pi b^{4} x^{3} - \pi a b^{3} x^{2} + \pi a^{2} b^{2} x - \pi a^{3} b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - {\left (3 \, b^{2} x - 5 \, a b\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{4 \, \pi ^{2} b^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{3} {\rm fresnels}\left (b x + a\right ), x\right ) \]