\[ \int \frac {S(b x)}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {b^{3} \pi \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 x}-\frac {b^{4} \pi ^{2} \mathrm {S}\left (b x \right )}{12}-\frac {\mathrm {S}\left (b x \right )}{4 x^{4}}-\frac {b \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 x^{3}} \]
command
integrate(fresnel_sin(b*x)/x^5,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {3}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{4}}{64 \, x^{3}} - \frac {\operatorname {S}\left (b x\right )}{4 \, x^{4}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{5}}\,{d x} \]________________________________________________________________________________________