Performance of discriminant calculations over power series rings

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Let $P=\mathbb{Q}[t_1,\ldots ,t_n]$ be the ring of polynomials of variables $t_1,\ldots ,t_n$ with rational coefficients, and let $S=P[[q]]$ be the ring of power series of variable $q$ with coefficients in $P$. Given a polynomial $p\in S[x]$ of variable $x$ with coefficients in $S$, denote $\Delta(p)=\operatorname{resultant}(p(x),p'(x))\in S$ the discriminant of $p$, which is the resultant of $p(x)$ and its $x$-derivative $p'(x)$. What is the performance of SageMath in computing $\Delta(p)$ modulo $q^A$ for a given $A\in\mathbb{N}$?

I'm interested in a theoretical answer describing the complexity class of the algorithms used by SageMath, as well as in an empirical answer estimating how much time it would take to do this calculation in the case below, where $n=4$, $A=2$, and $p(x)$ is as shown. It doesn't seem to terminate in any reasonable time on a laptop.

P.<t2,t3,t4,t5> = PolynomialRing(QQ)
S.<q> = PowerSeriesRing(R, default_prec=2)
p = x^6 + ((-1/6*t2^3 - t2^2*t3 - 1/2*t2*t3^2 - 2*t2*t4 - t3*t4 - 2*t5)*q)*x^5 + ((28*t2*t3 - 32*t4*t5)*q + (1/6*t2^5*t3 - 1/6*t2^4*t3^2 + 1/2*t2^3*t3^3 + 1/3*t2^4*t4 + 5/3*t2^3*t3*t4 + 2*t2^2*t3^2*t4 + 3/2*t2^2*t4^2 + 2*t2*t3*t4^2 + 1/3*t2^3*t5 - 6*t2^2*t3*t5 + t2*t3^2*t5 + 9*t2*t4*t5^2 + 3*t2*t4*t5 + 2*t3*t4*t5)*q^2)*x^4 + ((-2/3*t2^4*t3 - 104/3*t2^3*t3^2 - 8*t2^2*t3^3 + 16/3*t2^3*t4*t5 + 40*t2^2*t3*t4*t5 + 16*t2*t3^2*t4*t5 + 18*t2^3*t4 - 44*t2^2*t3*t4 + 8*t2*t3^2*t4 + 96*t2*t4^2*t5 + 32*t3*t4^2*t5 + 48*t2*t4^2 + 24*t3*t4^2 + 33*t2^2*t5 - 152*t2*t3*t5 + 33*t3^2*t5 + 224*t4*t5^2 + 80*t4*t5)*q^2 + (-1/12*t2^7*t3^2 + 1/6*t2^6*t3^3 - 1/3*t2^5*t3^4 + 1/12*t2^6*t3*t4 + 7/6*t2^5*t3^2*t4 + 1/12*t2^4*t3^3*t4 + 1/12*t2^5*t4^2 + 8/3*t2^4*t3*t4^2 + 23/12*t2^3*t3^2*t4^2 - 7/6*t2^5*t3*t5 + 4/3*t2^4*t3^2*t5 - 11/2*t2^3*t3^3*t5 + 3/2*t2^4*t4*t5^2 + 15/2*t2^2*t3^2*t4*t5^2 + 2*t2^3*t4^3 + 7/2*t2^2*t3*t4^3 + 1/6*t2^4*t4*t5 + 17/3*t2^3*t3*t4*t5 - 19/2*t2^2*t3^2*t4*t5 + 15*t2*t3*t4^2*t5^2 + 2*t2*t4^4 + 6*t2^2*t4^2*t5 - 5*t2*t3*t4^2*t5 - 1/6*t2^3*t5^2 + 3*t2^2*t3*t5^2 - 1/2*t2*t3^2*t5^2 + 12*t4^3*t5^2 + 4*t4^3*t5 + 6*t2*t4*t5^2 - t3*t4*t5^2 + 2*t5^3)*q^3)*x^3 + ((-9*t2^4 + 16*t2^3*t3 + 178*t2^2*t3^2 + 16*t2*t3^3 - 9*t3^4 - 32*t2^2*t4*t5 - 448*t2*t3*t4*t5 - 32*t3^2*t4*t5 + 256*t4^2*t5^2 + 128*t2^2*t4 - 256*t2*t3*t4 + 128*t3^2*t4 + 512*t4^2*t5 + 256*t4^2)*q^2 + (-26/9*t2^6*t3^2 + 16/3*t2^5*t3^3 - 2/3*t2^4*t3^4 - 4/3*t2^5*t3*t4*t5 - 20/3*t2^4*t3^2*t4*t5 - 4*t2^3*t3^3*t4*t5 - 10*t2^5*t3*t4 + 24*t2^4*t3^2*t4 - 22/3*t2^3*t3^3*t4 + 8*t2^4*t4^2*t5 - 16/3*t2^3*t3*t4^2*t5 + 24*t2^2*t3^2*t4^2*t5 - 8*t2^4*t4^2 + 18*t2^3*t3*t4^2 - 16*t2^2*t3^2*t4^2 - 311/3*t2^4*t3*t5 + 57*t2^3*t3^2*t5 - 145*t2^2*t3^3*t5 + 16*t2^2*t4^3*t5 + 64*t2*t3*t4^3*t5 + 344/3*t2^3*t4*t5^2 - 148*t2^2*t3*t4*t5^2 + 192*t2*t3^2*t4*t5^2 + 144*t2*t4^2*t5^3 + 8*t2^2*t4^3 - 118/3*t2^3*t4*t5 + 23*t2^2*t3*t4*t5 - 50*t2*t3^2*t4*t5 + 64*t4^4*t5 + 48*t2*t4^2*t5^2 + 80*t3*t4^2*t5^2 + 32*t4^4 - 36*t2*t4^2*t5 + 24*t3*t4^2*t5 - 66*t2^2*t5^2 + 194*t2*t3*t5^2 - 66*t3^2*t5^2 - 208*t4*t5^3 - 96*t4*t5^2)*q^3 + (-5/72*t2^8*t3^4 - 11/36*t2^7*t3^3*t4 - 19/36*t2^6*t3^2*t4^2 - 7/6*t2^6*t3^3*t5 + t2^5*t3^2*t4*t5^2 - 13/12*t2^5*t3*t4^3 - 4*t2^5*t3^2*t4*t5 + 7/2*t2^4*t3*t4^2*t5^2 - 5/6*t2^4*t4^4 - 17/3*t2^4*t3*t4^2*t5 - 19/6*t2^4*t3^2*t5^2 + t2^3*t4^3*t5^2 + 3*t2^3*t3*t4*t5^3 - 14/3*t2^3*t4^3*t5 - 7/3*t2^3*t3*t4*t5^2 - 6*t2^2*t4^2*t5^3 - 19/2*t2^2*t4^2*t5^2 + 4*t2^2*t3*t5^3 - 9*t2*t4*t5^4 - 7*t2*t4*t5^3 - t5^4)*q^4)*x^2 + ((55/3*t2^6*t3 - 28*t2^5*t3^2 - 198*t2^4*t3^3 - 196/3*t2^3*t3^4 + 25*t2^2*t3^5 - 56/3*t2^5*t4*t5 + 224/3*t2^4*t3*t4*t5 + 520*t2^3*t3^2*t4*t5 + 192*t2^2*t3^3*t4*t5 - 32*t2*t3^4*t4*t5 - 272/3*t2^3*t4^2*t5^2 - 384*t2^2*t3*t4^2*t5^2 - 176*t2*t3^2*t4^2*t5^2 + 26/3*t2^5*t4 - 52/3*t2^4*t3*t4 + 120*t2^3*t3^2*t4 - 68*t2^2*t3^3*t4 + 10*t2*t3^4*t4 - 16/3*t2^3*t4^2*t5 + 144*t2^2*t3*t4^2*t5 + 368*t2*t3^2*t4^2*t5 - 16*t3^3*t4^2*t5 - 256*t2*t4^3*t5^2 - 256*t3*t4^3*t5^2 - 272/3*t2^3*t4^2 + 376*t2^2*t3*t4^2 + 224*t2*t3^2*t4^2 - 8*t3^3*t4^2 + 18*t2^4*t5 - 288*t2^3*t3*t5 - 484*t2^2*t3^2*t5 - 288*t2*t3^3*t5 + 18*t3^4*t5 - 384*t2*t4^3*t5 - 128*t3*t4^3*t5 + 416*t2^2*t4*t5^2 + 704*t2*t3*t4*t5^2 + 416*t3^2*t4*t5^2 - 128*t2*t4^3 + 128*t3*t4^3 + 32*t2^2*t4*t5 + 448*t2*t3*t4*t5 + 32*t3^2*t4*t5)*q^3 + (17/18*t2^8*t3^3 + 4/9*t2^7*t3^4 + 1/3*t2^6*t3^5 - 2/3*t2^7*t3^2*t4*t5 - 8/3*t2^6*t3^3*t4*t5 - 2*t2^5*t3^4*t4*t5 + 40/9*t2^7*t3^2*t4 + 44/9*t2^6*t3^3*t4 + 22/3*t2^5*t3^4*t4 - 4*t2^6*t3*t4^2*t5 - 40/3*t2^5*t3^2*t4^2*t5 - 12*t2^4*t3^3*t4^2*t5 + 25/3*t2^6*t3*t4^2 + 40/3*t2^5*t3^2*t4^2 + 58/3*t2^4*t3^3*t4^2 + 100/3*t2^6*t3^2*t5 - 39*t2^5*t3^3*t5 + 166/3*t2^4*t3^4*t5 - 16/3*t2^5*t4^3*t5 - 24*t2^4*t3*t4^3*t5 - 24*t2^3*t3^2*t4^3*t5 - 184/3*t2^5*t3*t4*t5^2 + 128/3*t2^4*t3^2*t4*t5^2 - 156*t2^3*t3^3*t4*t5^2 + 24*t2^4*t4^2*t5^3 + 120*t2^2*t3^2*t4^2*t5^3 + 11/3*t2^5*t4^3 + 16/3*t2^4*t3*t4^3 + 10*t2^3*t3^2*t4^3 + 359/6*t2^5*t3*t4*t5 - 349/3*t2^4*t3^2*t4*t5 + 251/6*t2^3*t3^3*t4*t5 - 80/3*t2^3*t4^4*t5 - 16*t2^2*t3*t4^4*t5 - 130/3*t2^4*t4^2*t5^2 + 448/3*t2^3*t3*t4^2*t5^2 - 220*t2^2*t3^2*t4^2*t5^2 + 240*t2*t3*t4^3*t5^3 - 40/3*t2^3*t4^4 - 4*t2^2*t3*t4^4 + 61/2*t2^4*t4^2*t5 - 38/3*t2^3*t3*t4^2*t5 - 239/2*t2^2*t3^2*t4^2*t5 + 13*t2^4*t3*t5^2 - 556/3*t2^3*t3^2*t5^2 + 163*t2^2*t3^3*t5^2 - 32*t2^2*t4^3*t5^2 - 8*t2*t3*t4^3*t5^2 + 17*t2^3*t4*t5^3 + 192*t2^2*t3*t4*t5^3 - 223*t2*t3^2*t4*t5^3 + 192*t4^4*t5^3 - 32*t2^2*t4^3*t5 - 124*t2*t3*t4^3*t5 + 67*t2^3*t4*t5^2 + 14*t2^2*t3*t4*t5^2 + 229*t2*t3^2*t4*t5^2 + 192*t4^4*t5^2 - 240*t2*t4^2*t5^3 - 400*t3*t4^2*t5^3 + 48*t4^4*t5 - 104*t2*t4^2*t5^2 - 144*t3*t4^2*t5^2 + 33*t2^2*t5^3 + 20*t2*t3*t5^3 + 33*t3^2*t5^3 - 224*t4*t5^4 - 80*t4*t5^3)*q^4)*x + (-76/9*t2^8*t3^2 - 10*t2^7*t3^3 + 4*t2^6*t3^4 - 22*t2^5*t3^5 - 64/3*t2^4*t3^6 + 52/3*t2^7*t3*t4*t5 + 112/3*t2^6*t3^2*t4*t5 + 20/3*t2^5*t3^3*t4*t5 + 64*t2^4*t3^4*t4*t5 + 56*t2^3*t3^5*t4*t5 - 8*t2^6*t4^2*t5^2 - 64/3*t2^5*t3*t4^2*t5^2 - 80/3*t2^4*t3^2*t4^2*t5^2 - 64*t2^3*t3^3*t4^2*t5^2 - 40*t2^2*t3^4*t4^2*t5^2 - 25/2*t2^7*t3*t4 - 130/3*t2^6*t3^2*t4 - 235/3*t2^5*t3^3*t4 - 58*t2^4*t3^4*t4 - 103/6*t2^3*t3^5*t4 + 32/3*t2^6*t4^2*t5 + 400/3*t2^5*t3*t4^2*t5 + 272*t2^4*t3^2*t4^2*t5 + 256*t2^3*t3^3*t4^2*t5 + 80*t2^2*t3^4*t4^2*t5 - 256/3*t2^4*t4^3*t5^2 - 736/3*t2^3*t3*t4^3*t5^2 - 256*t2^2*t3^2*t4^3*t5^2 - 80*t2*t3^3*t4^3*t5^2 - 9/2*t2^6*t4^2 + 28/3*t2^5*t3*t4^2 - 337/3*t2^4*t3^2*t4^2 + 36*t2^3*t3^3*t4^2 + 79/2*t2^2*t3^4*t4^2 - 7*t2^6*t3*t5 - 554/3*t2^4*t3^3*t5 - 240*t2^3*t3^4*t5 - 51*t2^2*t3^5*t5 + 16*t2^4*t4^3*t5 + 976/3*t2^3*t3*t4^3*t5 + 160*t2^2*t3^2*t4^3*t5 + 5/3*t2^5*t4*t5^2 + 88*t2^4*t3*t4*t5^2 + 482*t2^3*t3^2*t4*t5^2 + 520*t2^2*t3^3*t4*t5^2 + 71*t2*t3^4*t4*t5^2 - 256*t2^2*t4^4*t5^2 - 256*t2*t3*t4^4*t5^2 - 64*t3^2*t4^4*t5^2 - 128*t2^3*t4^2*t5^3 - 352*t2^2*t3*t4^2*t5^3 - 256*t2*t3^2*t4^2*t5^3 + 88/3*t2^4*t4^3 + 92/3*t2^3*t3*t4^3 + 112*t2^2*t3^2*t4^3 + 36*t2*t3^3*t4^3 - 43/3*t2^5*t4*t5 + 68*t2^4*t3*t4*t5 - 362*t2^3*t3^2*t4*t5 - 196*t2^2*t3^3*t4*t5 - 61*t2*t3^4*t4*t5 - 128*t2^2*t4^4*t5 - 192*t2*t3*t4^4*t5 - 64*t3^2*t4^4*t5 + 944*t2^2*t3*t4^2*t5^2 + 544*t2*t3^2*t4^2*t5^2 + 112*t3^3*t4^2*t5^2 - 512*t2*t4^3*t5^3 - 256*t3*t4^3*t5^3 - 16*t2^2*t4^4 - 32*t2*t3*t4^4 - 16*t3^2*t4^4 + 80*t2^3*t4^2*t5 + 296*t2^2*t3*t4^2*t5 + 256*t2*t3^2*t4^2*t5 + 40*t3^3*t4^2*t5 - 9*t2^4*t5^2 - 240*t2^3*t3*t5^2 - 718*t2^2*t3^2*t5^2 - 240*t2*t3^3*t5^2 - 9*t3^4*t5^2 - 640*t2*t4^3*t5^2 - 384*t3*t4^3*t5^2 + 384*t2^2*t4*t5^3 + 1280*t2*t3*t4*t5^3 + 384*t3^2*t4*t5^3 - 256*t4^2*t5^4 - 128*t2*t4^3*t5 - 128*t3*t4^3*t5 + 96*t2^2*t4*t5^2 + 320*t2*t3*t4*t5^2 + 96*t3^2*t4*t5^2 - 512*t4^2*t5^3 - 256*t4^2*t5^2)*q^4 + (157/12*t2^7*t3^4*t5 - 32*t2^6*t3^3*t4*t5^2 + 16*t2^5*t3^2*t4^2*t5^3 + 185/3*t2^6*t3^3*t4*t5 - 128*t2^5*t3^2*t4^2*t5^2 + 56*t2^4*t3*t4^3*t5^3 + 129/2*t2^5*t3^2*t4^2*t5 + 53*t2^5*t3^3*t5^2 - 96*t2^4*t3*t4^3*t5^2 - 98*t2^4*t3^2*t4*t5^3 + 16*t2^3*t4^4*t5^3 + 48*t2^3*t3*t4^2*t5^4 + 27/2*t2^4*t3*t4^3*t5 + 50*t2^4*t3^2*t4*t5^2 - 32*t2^3*t4^4*t5^2 + 47*t2^3*t3*t4^2*t5^3 - 96*t2^2*t4^3*t5^4 - 14*t2^3*t4^4*t5 + 50*t2^3*t3*t4^2*t5^2 + 71*t2^3*t3^2*t5^3 - 60*t2^2*t4^3*t5^3 + 54*t2^2*t3*t4*t5^4 - 144*t2*t4^2*t5^5 - 36*t2^2*t4^3*t5^2 + 213*t2^2*t3*t4*t5^3 - 144*t2*t4^2*t5^4 + 12*t2*t4^2*t5^3 - 90*t2*t3*t5^4 + 240*t4*t5^5 + 96*t4*t5^4)*q^5
Delta = p.resultant(p.differentiate())