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Access the PathInt command through the Context Panel

$x y z$$\stackrel{\text{line integral}}{\to }$${\∫}_0^{1}3\left(1\+4t\right)\left(2\+t\right)\left(3-t\right)\sqrt{2}\ⅆt$$\=$$\frac{111}{2}\sqrt{2}$

Form and evaluate the line integral via the PathInt command

$\mathrm{PathInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Line}\left(⟨1\,2\,3⟩\,⟨5\,3\,2⟩\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{1}3\left(1\+4t\right)\left(2\+t\right)\left(3-t\right)\sqrt{2}\ⅆt$

$\mathrm{PathInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Line}\left(⟨1\,2\,3⟩\,⟨5\,3\,2⟩\right)\right)$ = $\frac{111}{2}\sqrt{2}$$$

Obtain a parametric representation of the line segment

${\mathbf{P}}_1\,{\mathbf{P}}_2≔⟨1\,2\,3⟩\,⟨5\,3\,2⟩\:$

$\mathbf{R}≔{\mathbf{P}}_1\+t \left({\mathbf{P}}_2-{\mathbf{P}}_1\right)\:$$$

Obtain $\mathrm{\ρ}\=\sqrt{{\stackrel{\.}{x}}^2\+{\stackrel{\.}{y}}^2\+{\stackrel{\.}{z}}^2}\=∥\stackrel{\.}{\mathbf{R}}∥$ 

$∥\frac{\ⅆ}{\ⅆ t} \mathbf{R}∥$ = $3\sqrt{2}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\ρ}$$$

Form the integrand $f\left(x\left(t\right)\,y\left(t\right)\,z\left(t\right)\right)\cdot \mathrm{\ρ}$ and integrate with respect to $t$ 

$\left(\genfrac{}{}{0ex}{}{x y z}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\={\mathbf{R}}_1\,y\={\mathbf{R}}_2\,z\={\mathbf{R}}_3}\right)\cdot \mathrm{\ρ}$

$3\left(1\+4t\right)\left(2\+t\right)\left(3-t\right)\sqrt{2}$

$\stackrel{\text{integrate w.r.t. t}}{\to }$

${\∫}_0^{1}3\left(1\+4t\right)\left(2\+t\right)\left(3-t\right)\sqrt{2}\ⅆt$

$\=$

$\frac{111}{2}\sqrt{2}$