\[ \int x \text {Si}(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {a \cos \left (b x +a \right )}{2 b^{2}}+\frac {x \cos \left (b x +a \right )}{2 b}-\frac {a^{2} \sinIntegral \left (b x +a \right )}{2 b^{2}}+\frac {x^{2} \sinIntegral \left (b x +a \right )}{2}-\frac {\sin \left (b x +a \right )}{2 b^{2}} \]
command
integrate(x*sin_integral(b*x+a),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{2} \, x^{2} \operatorname {Si}\left (b x + a\right ) - \frac {a^{2} {\left (-i \, {\rm Ei}\left (i \, b x + i \, a\right ) + i \, {\rm Ei}\left (-i \, b x - i \, a\right )\right )} - 2 \, {\left (b x - a\right )} \cos \left (b x + a\right ) + 2 \, \sin \left (b x + a\right )}{4 \, b^{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int x {\rm Si}\left (b x + a\right )\,{d x} \]________________________________________________________________________________________