In [C. Bujac, J. Llibre, N. Vulpe, Qual. Theory Dyn. Syst. 15(2016), 327-348] all first integrals and phase portraits were constructed for the family of cubic differential systems with the maximum number of invariant straight lines, i.e. 9 (considered with their multiplicities). Here we continue this investigation for systems with invariant straight lines of total multiplicity eight. For such systems the classification according to the configurations of invariant lines in terms of affine invariant polynomials was done in [C. Bujace, Bul. Acad. Ştiinţe Repub. Mold Repub. Mold. Mat. 75(2014),102-105],[C. Bujac, N. Vulpe, J. Math. Anal. Appl. 423(2015), 1025-1080], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst. 14(2015), 109-137], [C. Bujac, N. Vulpe, Electron. J. Qual. Theory Differ. Equ. 2015, No. 74, 1-38], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst. 16(2017), 1-30] and all possible 51 configurations were constructed. In this article we prove that all systems in this class are integrable. For each one of the 51 such classes we compute the corresponding first integral and we draw the corresponding phase portrait.