\[ \int x^5 \text {FresnelC}(b x) \, dx \]
Optimal antiderivative \[ -\frac {5 x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 b^{3} \pi ^{2}}+\frac {x^{6} \FresnelC \left (b x \right )}{6}-\frac {5 \,\mathrm {S}\left (b x \right )}{2 b^{6} \pi ^{3}}+\frac {5 x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{5} \pi ^{3}}-\frac {x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 b \pi } \]
command
integrate(x^5*fresnel_cos(b*x),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\pi ^{3} b^{7} x^{6} \operatorname {C}\left (b x\right ) - 5 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{6} x^{5} - 15 \, b^{2} x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 15 \, \sqrt {b^{2}} \operatorname {S}\left (\sqrt {b^{2}} x\right )}{6 \, \pi ^{3} b^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{5} {\rm fresnelc}\left (b x\right ), x\right ) \]