3.30.72 $\int \frac{-81+x-7776x^5+96x^6+64x^5\log (-81+x)+(-12960x^4+160x^5){\log }^2(-81+x)+96x^4{\log }^3(-81+x)+(-7776x^3+96x^4){\log }^4(-81+x)+48x^3{\log }^5(-81+x)+(-1944x^2+24x^3){\log }^6(-81+x)+8x^2{\log }^7(-81+x)+(-162x+2x^2){\log }^8(-81+x)}{-81+x}dx$

Optimal. Leaf size=18 $$x+x^2{(2x+{\log }^2(-81+x))}^4$$

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Rubi [B]  time = 2.32, antiderivative size = 170, normalized size of antiderivative = 9.44, number of steps used = 187, number of rules used = 17, integrand size = 131, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.130, Rules used = {6742, 2411, 43, 2334, 2301, 2398, 2346, 2302, 30, 2296, 2295, 2330, 2305, 2304, 2401, 2389, 2390} $$\begin{array}{c}16x^6+32x^5{\log }^2(x-81)+x+(81-x{)}^2{\log }^8(x-81)-162(81-x){\log }^8(x-81)+6561{\log }^8(x-81)-8(81-x{)}^3{\log }^6(x-81)+1944(81-x{)}^2{\log }^6(x-81)-157464(81-x){\log }^6(x-81)+4251528{\log }^6(x-81)+24(81-x{)}^4{\log }^4(x-81)-7776(81-x{)}^3{\log }^4(x-81)+944784(81-x{)}^2{\log }^4(x-81)-51018336(81-x){\log }^4(x-81)[phone]\mathrm{}{\log }^4(x-81)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-81 + x - 7776*x^5 + 96*x^6 + 64*x^5*Log[-81 + x] + (-12960*x^4 + 160*x^5)*Log[-81 + x]^2 + 96*x^4*Log[-8
1 + x]^3 + (-7776*x^3 + 96*x^4)*Log[-81 + x]^4 + 48*x^3*Log[-81 + x]^5 + (-1944*x^2 + 24*x^3)*Log[-81 + x]^6 +
8*x^2*Log[-81 + x]^7 + (-162*x + 2*x^2)*Log[-81 + x]^8)/(-81 + x),x]

[Out]

x + 16*x^6 + 32*x^5*Log[-81 + x]^2 + 1033121304*Log[-81 + x]^4 - 51018336*(81 - x)*Log[-81 + x]^4 + 944784*(81
- x)^2*Log[-81 + x]^4 - 7776*(81 - x)^3*Log[-81 + x]^4 + 24*(81 - x)^4*Log[-81 + x]^4 + 4251528*Log[-81 + x]^
6 - 157464*(81 - x)*Log[-81 + x]^6 + 1944*(81 - x)^2*Log[-81 + x]^6 - 8*(81 - x)^3*Log[-81 + x]^6 + 6561*Log[-
81 + x]^8 - 162*(81 - x)*Log[-81 + x]^8 + (81 - x)^2*Log[-81 + x]^8

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (1+96x^5+\frac{64x^5\log (-81+x)}{-81+x}+160x^4{\log }^2(-81+x)+\frac{96x^4{\log }^3(-81+x)}{-81+x}+96x^3{\log }^4(-81+x)+\frac{48x^3{\log }^5(-81+x)}{-81+x}+24x^2{\log }^6(-81+x)+\frac{8x^2{\log }^7(-81+x)}{-81+x}+2x{\log }^8(-81+x))dx\\ & =x+16x^6+2\int x{\log }^8(-81+x)dx+8\int \frac{x^2{\log }^7(-81+x)}{-81+x}dx+24\int x^2{\log }^6(-81+x)dx+48\int \frac{x^3{\log }^5(-81+x)}{-81+x}dx+64\int \frac{x^5\log (-81+x)}{-81+x}dx+96\int \frac{x^4{\log }^3(-81+x)}{-81+x}dx+96\int x^3{\log }^4(-81+x)dx+160\int x^4{\log }^2(-81+x)dx\\ & =x+16x^6+32x^5{\log }^2(-81+x)+2\int (81{\log }^8(-81+x)+(-81+x){\log }^8(-81+x))dx+8\mathrm{Subst}(\int \frac{(81+x{)}^2{\log }^7(x)}{x}dx,x,-81+x)+24\int (6561{\log }^6(-81+x)+162(-81+x){\log }^6(-81+x)+(-81+x{)}^2{\log }^6(-81+x))dx+48\mathrm{Subst}(\int \frac{(81+x{)}^3{\log }^5(x)}{x}dx,x,-81+x)-64\int \frac{x^5\log (-81+x)}{-81+x}dx+64\mathrm{Subst}(\int \frac{(81+x{)}^5\log (x)}{x}dx,x,-81+x)+96\int (531441{\log }^4(-81+x)+19683(-81+x){\log }^4(-81+x)+243(-81+x{)}^2{\log }^4(-81+x)+(-81+x{)}^3{\log }^4(-81+x))dx+96\mathrm{Subst}(\int \frac{(81+x{)}^4{\log }^3(x)}{x}dx,x,-81+x)\\ & =x+16x^6-\frac{16}{5}(4304672100(81-x)-53144100(81-x{)}^2+437400(81-x{)}^3-2025(81-x{)}^4+4(81-x{)}^5-69735688020\log (-81+x))\log (-81+x)+32x^5{\log }^2(-81+x)+2\int (-81+x){\log }^8(-81+x)dx+8\mathrm{Subst}(\int (81+x){\log }^7(x)dx,x,-81+x)+24\int (-81+x{)}^2{\log }^6(-81+x)dx+48\mathrm{Subst}(\int (81+x{)}^2{\log }^5(x)dx,x,-81+x)-64\mathrm{Subst}(\int \frac{(81+x{)}^5\log (x)}{x}dx,x,-81+x)-64\mathrm{Subst}(\int (215233605+2657205x+21870x^2+\frac{405x^3}{4}+\frac{x^4}{5}+\frac{3486784401\log (x)}{x})dx,x,-81+x)+96\int (-81+x{)}^3{\log }^4(-81+x)dx+96\mathrm{Subst}(\int (81+x{)}^3{\log }^3(x)dx,x,-81+x)+162\int {\log }^8(-81+x)dx+648\mathrm{Subst}(\int \frac{(81+x){\log }^7(x)}{x}dx,x,-81+x)+3888\int (-81+x){\log }^6(-81+x)dx+3888\mathrm{Subst}(\int \frac{(81+x{)}^2{\log }^5(x)}{x}dx,x,-81+x)+7776\mathrm{Subst}(\int \frac{(81+x{)}^3{\log }^3(x)}{x}dx,x,-81+x)+23328\int (-81+x{)}^2{\log }^4(-81+x)dx+157464\int {\log }^6(-81+x)dx+1889568\int (-81+x){\log }^4(-81+x)dx+51018336\int {\log }^4(-81+x)dx\\ & =-85030560(81-x{)}^2+466560(81-x{)}^3-1620(81-x{)}^4+\frac{64}{25}(81-x{)}^5-13774950719x+16x^6+32x^5{\log }^2(-81+x)+2\mathrm{Subst}(\int x{\log }^8(x)dx,x,-81+x)+8\mathrm{Subst}(\int (81{\log }^7(x)+x{\log }^7(x))dx,x,-81+x)+24\mathrm{Subst}(\int x^2{\log }^6(x)dx,x,-81+x)+48\mathrm{Subst}(\int (6561{\log }^5(x)+162x{\log }^5(x)+x^2{\log }^5(x))dx,x,-81+x)+64\mathrm{Subst}(\int (215233605+2657205x+21870x^2+\frac{405x^3}{4}+\frac{x^4}{5}+\frac{3486784401\log (x)}{x})dx,x,-81+x)+96\mathrm{Subst}(\int x^3{\log }^4(x)dx,x,-81+x)+96\mathrm{Subst}(\int (531441{\log }^3(x)+19683x{\log }^3(x)+243x^2{\log }^3(x)+x^3{\log }^3(x))dx,x,-81+x)+162\mathrm{Subst}(\int {\log }^8(x)dx,x,-81+x)+648\mathrm{Subst}(\int {\log }^7(x)dx,x,-81+x)+3888\mathrm{Subst}(\int (81+x){\log }^5(x)dx,x,-81+x)+3888\mathrm{Subst}(\int x{\log }^6(x)dx,x,-81+x)+7776\mathrm{Subst}(\int (81+x{)}^2{\log }^3(x)dx,x,-81+x)+23328\mathrm{Subst}(\int x^2{\log }^4(x)dx,x,-81+x)+52488\mathrm{Subst}(\int \frac{{\log }^7(x)}{x}dx,x,-81+x)+157464\mathrm{Subst}(\int {\log }^6(x)dx,x,-81+x)+314928\mathrm{Subst}(\int \frac{(81+x){\log }^5(x)}{x}dx,x,-81+x)+629856\mathrm{Subst}(\int \frac{(81+x{)}^2{\log }^3(x)}{x}dx,x,-81+x)+1889568\mathrm{Subst}(\int x{\log }^4(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int {\log }^4(x)dx,x,-81+x)-223154201664\mathrm{Subst}(\int \frac{\log (x)}{x}dx,x,-81+x)\\ & =x+16x^6-111577100832{\log }^2(-81+x)+32x^5{\log }^2(-81+x)-51018336(81-x){\log }^4(-81+x)+944784(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)-157464(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)-648(81-x){\log }^7(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)+648\mathrm{Subst}(\int {\log }^7(x)dx,x,-81+x)-1296\mathrm{Subst}(\int {\log }^7(x)dx,x,-81+x)+3888\mathrm{Subst}(\int (81{\log }^5(x)+x{\log }^5(x))dx,x,-81+x)-4536\mathrm{Subst}(\int {\log }^6(x)dx,x,-81+x)+7776\mathrm{Subst}(\int x{\log }^5(x)dx,x,-81+x)+7776\mathrm{Subst}(\int (6561{\log }^3(x)+162x{\log }^3(x)+x^2{\log }^3(x))dx,x,-81+x)-11664\mathrm{Subst}(\int x{\log }^5(x)dx,x,-81+x)+23328\mathrm{Subst}(\int x^2{\log }^3(x)dx,x,-81+x)-31104\mathrm{Subst}(\int x^2{\log }^3(x)dx,x,-81+x)+52488\mathrm{Subst}(\int x^{7}dx,x,\log (-81+x))+2\left(314928\mathrm{Subst}(\int {\log }^5(x)dx,x,-81+x)\right)+629856\mathrm{Subst}(\int (81+x){\log }^3(x)dx,x,-81+x)-944784\mathrm{Subst}(\int {\log }^5(x)dx,x,-81+x)+1889568\mathrm{Subst}(\int x{\log }^3(x)dx,x,-81+x)-3779136\mathrm{Subst}(\int x{\log }^3(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \frac{{\log }^5(x)}{x}dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \frac{(81+x){\log }^3(x)}{x}dx,x,-81+x)-204073344\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \frac{\log (x)}{x}dx,x,-81+x)\\ & =x+16x^6+32x^5{\log }^2(-81+x)[phone]\mathrm{}(81-x){\log }^3(-81+x)-944784(81-x{)}^2{\log }^3(-81+x)+2592(81-x{)}^3{\log }^3(-81+x)-51018336(81-x){\log }^4(-81+x)+944784(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)+944784(81-x){\log }^5(-81+x)-1944(81-x{)}^2{\log }^5(-81+x)-152928(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)+6561{\log }^8(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)+3888\mathrm{Subst}(\int x{\log }^5(x)dx,x,-81+x)-4536\mathrm{Subst}(\int {\log }^6(x)dx,x,-81+x)+7776\mathrm{Subst}(\int x^2{\log }^3(x)dx,x,-81+x)+9072\mathrm{Subst}(\int {\log }^6(x)dx,x,-81+x)-19440\mathrm{Subst}(\int x{\log }^4(x)dx,x,-81+x)-23328\mathrm{Subst}(\int x^2{\log }^2(x)dx,x,-81+x)+27216\mathrm{Subst}(\int {\log }^5(x)dx,x,-81+x)+29160\mathrm{Subst}(\int x{\log }^4(x)dx,x,-81+x)+31104\mathrm{Subst}(\int x^2{\log }^2(x)dx,x,-81+x)+314928\mathrm{Subst}(\int {\log }^5(x)dx,x,-81+x)+629856\mathrm{Subst}(\int (81{\log }^3(x)+x{\log }^3(x))dx,x,-81+x)+1259712\mathrm{Subst}(\int x{\log }^3(x)dx,x,-81+x)+2(-314928(81-x){\log }^5(-81+x)-1574640\mathrm{Subst}(\int {\log }^4(x)dx,x,-81+x))-2834352\mathrm{Subst}(\int x{\log }^2(x)dx,x,-81+x)+4723920\mathrm{Subst}(\int {\log }^4(x)dx,x,-81+x)+5668704\mathrm{Subst}(\int x{\log }^2(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int x^{5}dx,x,\log (-81+x))[phone]\mathrm{}\mathrm{}\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x))-153055008\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \frac{{\log }^3(x)}{x}dx,x,-81+x)\\ & =x+16x^6-459165024(81-x){\log }^2(-81+x)+1417176(81-x{)}^2{\log }^2(-81+x)-2592(81-x{)}^3{\log }^2(-81+x)+32x^5{\log }^2(-81+x)[phone]\mathrm{}(81-x){\log }^3(-81+x)-314928(81-x{)}^2{\log }^3(-81+x)-55742256(81-x){\log }^4(-81+x)+949644(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)+602640(81-x){\log }^5(-81+x)+4251528{\log }^6(-81+x)-157464(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)+6561{\log }^8(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)-7776\mathrm{Subst}(\int x^2{\log }^2(x)dx,x,-81+x)-9720\mathrm{Subst}(\int x{\log }^4(x)dx,x,-81+x)+15552\mathrm{Subst}(\int x^2\log (x)dx,x,-81+x)-20736\mathrm{Subst}(\int x^2\log (x)dx,x,-81+x)+27216\mathrm{Subst}(\int {\log }^5(x)dx,x,-81+x)+38880\mathrm{Subst}(\int x{\log }^3(x)dx,x,-81+x)-54432\mathrm{Subst}(\int {\log }^5(x)dx,x,-81+x)-58320\mathrm{Subst}(\int x{\log }^3(x)dx,x,-81+x)-136080\mathrm{Subst}(\int {\log }^4(x)dx,x,-81+x)+629856\mathrm{Subst}(\int x{\log }^3(x)dx,x,-81+x)-1574640\mathrm{Subst}(\int {\log }^4(x)dx,x,-81+x)-1889568\mathrm{Subst}(\int x{\log }^2(x)dx,x,-81+x)+2834352\mathrm{Subst}(\int x\log (x)dx,x,-81+x)-5668704\mathrm{Subst}(\int x\log (x)dx,x,-81+x)[phone]\mathrm{}\mathrm{}(81-x){\log }^4(-81+x)-314928(81-x){\log }^5(-81+x)+6298560\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x))-18895680\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)+2(-51018336(81-x){\log }^3(-81+x)-153055008\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x))[phone]\mathrm{}\mathrm{Subst}(\int \log (x)dx,x,-81+x)-1224440064\mathrm{Subst}(\int \log (x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int x^{3}dx,x,\log (-81+x))\\ & =708588(81-x{)}^2-576(81-x{)}^3+918330049x+16x^6+918330048(81-x)\log (-81+x)-1417176(81-x{)}^2\log (-81+x)+1728(81-x{)}^3\log (-81+x)-459165024(81-x){\log }^2(-81+x)+472392(81-x{)}^2{\log }^2(-81+x)+32x^5{\log }^2(-81+x)[phone]\mathrm{}(81-x){\log }^3(-81+x)-9720(81-x{)}^2{\log }^3(-81+x)[phone]\mathrm{}{\log }^4(-81+x)-54031536(81-x){\log }^4(-81+x)+944784(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)+629856(81-x){\log }^5(-81+x)+4251528{\log }^6(-81+x)-157464(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)+6561{\log }^8(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)+5184\mathrm{Subst}(\int x^2\log (x)dx,x,-81+x)+19440\mathrm{Subst}(\int x{\log }^3(x)dx,x,-81+x)-58320\mathrm{Subst}(\int x{\log }^2(x)dx,x,-81+x)+87480\mathrm{Subst}(\int x{\log }^2(x)dx,x,-81+x)-136080\mathrm{Subst}(\int {\log }^4(x)dx,x,-81+x)+272160\mathrm{Subst}(\int {\log }^4(x)dx,x,-81+x)+544320\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)-944784\mathrm{Subst}(\int x{\log }^2(x)dx,x,-81+x)+1889568\mathrm{Subst}(\int x\log (x)dx,x,-81+x)+6298560\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)+2(-6298560(81-x){\log }^3(-81+x)+1574640(81-x){\log }^4(-81+x)-314928(81-x){\log }^5(-81+x)-18895680\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x))[phone]\mathrm{}\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)-153055008\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{}(81-x){\log }^2(-81+x)-51018336(81-x){\log }^3(-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \log (x)dx,x,-81+x))\\ & =236196(81-x{)}^2+918330049x+16x^6+918330048(81-x)\log (-81+x)-472392(81-x{)}^2\log (-81+x)-362797056(81-x){\log }^2(-81+x)+14580(81-x{)}^2{\log }^2(-81+x)+32x^5{\log }^2(-81+x)[phone]\mathrm{}(81-x){\log }^3(-81+x)[phone]\mathrm{}{\log }^4(-81+x)-54167616(81-x){\log }^4(-81+x)+944784(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)+629856(81-x){\log }^5(-81+x)+4251528{\log }^6(-81+x)-157464(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)+6561{\log }^8(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)+2(-306110016x-306110016(81-x)\log (-81+x)[phone]\mathrm{}(81-x){\log }^2(-81+x)-51018336(81-x){\log }^3(-81+x))-29160\mathrm{Subst}(\int x{\log }^2(x)dx,x,-81+x)+58320\mathrm{Subst}(\int x\log (x)dx,x,-81+x)-87480\mathrm{Subst}(\int x\log (x)dx,x,-81+x)+544320\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)+944784\mathrm{Subst}(\int x\log (x)dx,x,-81+x)-1088640\mathrm{Subst}(\int {\log }^3(x)dx,x,-81+x)-1632960\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)-18895680\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{}(81-x){\log }^2(-81+x)-6298560(81-x){\log }^3(-81+x)+1574640(81-x){\log }^4(-81+x)-314928(81-x){\log }^5(-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \log (x)dx,x,-81+x))-113374080\mathrm{Subst}(\int \log (x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \log (x)dx,x,-81+x)\\ & =7290(81-x{)}^2+725594113x+16x^6+725594112(81-x)\log (-81+x)-14580(81-x{)}^2\log (-81+x)-342268416(81-x){\log }^2(-81+x)+32x^5{\log }^2(-81+x)[phone]\mathrm{}(81-x){\log }^3(-81+x)[phone]\mathrm{}{\log }^4(-81+x)-54167616(81-x){\log }^4(-81+x)+944784(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)+629856(81-x){\log }^5(-81+x)+4251528{\log }^6(-81+x)-157464(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)+6561{\log }^8(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)+2(-306110016x-306110016(81-x)\log (-81+x)[phone]\mathrm{}(81-x){\log }^2(-81+x)-51018336(81-x){\log }^3(-81+x))+2(-37791360x-37791360(81-x)\log (-81+x)[phone]\mathrm{}(81-x){\log }^2(-81+x)-6298560(81-x){\log }^3(-81+x)+1574640(81-x){\log }^4(-81+x)-314928(81-x){\log }^5(-81+x))+29160\mathrm{Subst}(\int x\log (x)dx,x,-81+x)-1632960\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)+3265920\mathrm{Subst}(\int \log (x)dx,x,-81+x)+3265920\mathrm{Subst}(\int {\log }^2(x)dx,x,-81+x)[phone]\mathrm{}\mathrm{Subst}(\int \log (x)dx,x,-81+x)\\ & =684536833x+16x^6+684536832(81-x)\log (-81+x)-343901376(81-x){\log }^2(-81+x)+32x^5{\log }^2(-81+x)[phone]\mathrm{}(81-x){\log }^3(-81+x)[phone]\mathrm{}{\log }^4(-81+x)-54167616(81-x){\log }^4(-81+x)+944784(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)+629856(81-x){\log }^5(-81+x)+4251528{\log }^6(-81+x)-157464(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)+6561{\log }^8(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)+2(-306110016x-306110016(81-x)\log (-81+x)[phone]\mathrm{}(81-x){\log }^2(-81+x)-51018336(81-x){\log }^3(-81+x))+2(-37791360x-37791360(81-x)\log (-81+x)[phone]\mathrm{}(81-x){\log }^2(-81+x)-6298560(81-x){\log }^3(-81+x)+1574640(81-x){\log }^4(-81+x)-314928(81-x){\log }^5(-81+x))+3265920\mathrm{Subst}(\int \log (x)dx,x,-81+x)-6531840\mathrm{Subst}(\int \log (x)dx,x,-81+x)\\ & =687802753x+16x^6+687802752(81-x)\log (-81+x)-343901376(81-x){\log }^2(-81+x)+32x^5{\log }^2(-81+x)[phone]\mathrm{}(81-x){\log }^3(-81+x)[phone]\mathrm{}{\log }^4(-81+x)-54167616(81-x){\log }^4(-81+x)+944784(81-x{)}^2{\log }^4(-81+x)-7776(81-x{)}^3{\log }^4(-81+x)+24(81-x{)}^4{\log }^4(-81+x)+629856(81-x){\log }^5(-81+x)+4251528{\log }^6(-81+x)-157464(81-x){\log }^6(-81+x)+1944(81-x{)}^2{\log }^6(-81+x)-8(81-x{)}^3{\log }^6(-81+x)+6561{\log }^8(-81+x)-162(81-x){\log }^8(-81+x)+(81-x{)}^2{\log }^8(-81+x)+2(-306110016x-306110016(81-x)\log (-81+x)[phone]\mathrm{}(81-x){\log }^2(-81+x)-51018336(81-x){\log }^3(-81+x))+2(-37791360x-37791360(81-x)\log (-81+x)[phone]\mathrm{}(81-x){\log }^2(-81+x)-6298560(81-x){\log }^3(-81+x)+1574640(81-x){\log }^4(-81+x)-314928(81-x){\log }^5(-81+x))\end{array}\end{array}$$

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Mathematica [B]  time = 0.13, size = 51, normalized size = 2.83 $$\begin{array}{c}-4518872583777+x+16x^6+32x^5{\log }^2(-81+x)+24x^4{\log }^4(-81+x)+8x^3{\log }^6(-81+x)+x^2{\log }^8(-81+x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-81 + x - 7776*x^5 + 96*x^6 + 64*x^5*Log[-81 + x] + (-12960*x^4 + 160*x^5)*Log[-81 + x]^2 + 96*x^4*
Log[-81 + x]^3 + (-7776*x^3 + 96*x^4)*Log[-81 + x]^4 + 48*x^3*Log[-81 + x]^5 + (-1944*x^2 + 24*x^3)*Log[-81 +
x]^6 + 8*x^2*Log[-81 + x]^7 + (-162*x + 2*x^2)*Log[-81 + x]^8)/(-81 + x),x]

[Out]

-4518872583777 + x + 16*x^6 + 32*x^5*Log[-81 + x]^2 + 24*x^4*Log[-81 + x]^4 + 8*x^3*Log[-81 + x]^6 + x^2*Log[-
81 + x]^8

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fricas [B]  time = 0.66, size = 50, normalized size = 2.78 $$\begin{array}{c}{x}^2\log {(x-81)}^8+8x^3\log {(x-81)}^6+24x^4\log {(x-81)}^4+32x^5\log {(x-81)}^2+16x^6+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-162*x)*log(x-81)^8+8*x^2*log(x-81)^7+(24*x^3-1944*x^2)*log(x-81)^6+48*x^3*log(x-81)^5+(96*x^
4-7776*x^3)*log(x-81)^4+96*x^4*log(x-81)^3+(160*x^5-12960*x^4)*log(x-81)^2+64*x^5*log(x-81)+96*x^6-7776*x^5+x-
81)/(x-81),x, algorithm="fricas")

[Out]

x^2*log(x - 81)^8 + 8*x^3*log(x - 81)^6 + 24*x^4*log(x - 81)^4 + 32*x^5*log(x - 81)^2 + 16*x^6 + x

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giac [B]  time = 0.29, size = 50, normalized size = 2.78 $$\begin{array}{c}{x}^2\log {(x-81)}^8+8x^3\log {(x-81)}^6+24x^4\log {(x-81)}^4+32x^5\log {(x-81)}^2+16x^6+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-162*x)*log(x-81)^8+8*x^2*log(x-81)^7+(24*x^3-1944*x^2)*log(x-81)^6+48*x^3*log(x-81)^5+(96*x^
4-7776*x^3)*log(x-81)^4+96*x^4*log(x-81)^3+(160*x^5-12960*x^4)*log(x-81)^2+64*x^5*log(x-81)+96*x^6-7776*x^5+x-
81)/(x-81),x, algorithm="giac")

[Out]

x^2*log(x - 81)^8 + 8*x^3*log(x - 81)^6 + 24*x^4*log(x - 81)^4 + 32*x^5*log(x - 81)^2 + 16*x^6 + x

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maple [B]  time = 0.47, size = 51, normalized size = 2.83

method	result	size
risch	$\ln {(x-81)}^8{x}^2+8\ln {(x-81)}^6{x}^3+24\ln {(x-81)}^4{x}^4+32\ln {(x-81)}^2{x}^5+16x^6+x$	$51$
derivativedivides	$-27113235502257+334731302497x+111577100832\ln {(x-81)}^2+10331213040{(x-81)}^2+\ln {(x-81)}^8{(x-81)}^2+162\ln {(x-81)}^8(x-81)+8\ln {(x-81)}^6{(x-81)}^3+1944\ln {(x-81)}^6{(x-81)}^2+24\ln {(x-81)}^4{(x-81)}^4+157464\ln {(x-81)}^6(x-81)+7776\ln {(x-81)}^4{(x-81)}^3+32\ln {(x-81)}^2{(x-81)}^5+944784\ln {(x-81)}^4{(x-81)}^2+12960\ln {(x-81)}^2{(x-81)}^4+51018336\ln {(x-81)}^4(x-81)+2099520\ln {(x-81)}^2{(x-81)}^3+170061120\ln {(x-81)}^2{(x-81)}^2+6887475360\ln {(x-81)}^2(x-81)+6561\ln {(x-81)}^8+4251528\ln {(x-81)}^6+1033121304\ln {(x-81)}^4+16{(x-81)}^6+7776{(x-81)}^5+1574640{(x-81)}^4+170061120{(x-81)}^3$	$246$
default	$-27113235502257+334731302497x+111577100832\ln {(x-81)}^2+10331213040{(x-81)}^2+\ln {(x-81)}^8{(x-81)}^2+162\ln {(x-81)}^8(x-81)+8\ln {(x-81)}^6{(x-81)}^3+1944\ln {(x-81)}^6{(x-81)}^2+24\ln {(x-81)}^4{(x-81)}^4+157464\ln {(x-81)}^6(x-81)+7776\ln {(x-81)}^4{(x-81)}^3+32\ln {(x-81)}^2{(x-81)}^5+944784\ln {(x-81)}^4{(x-81)}^2+12960\ln {(x-81)}^2{(x-81)}^4+51018336\ln {(x-81)}^4(x-81)+2099520\ln {(x-81)}^2{(x-81)}^3+170061120\ln {(x-81)}^2{(x-81)}^2+6887475360\ln {(x-81)}^2(x-81)+6561\ln {(x-81)}^8+4251528\ln {(x-81)}^6+1033121304\ln {(x-81)}^4+16{(x-81)}^6+7776{(x-81)}^5+1574640{(x-81)}^4+170061120{(x-81)}^3$	$246$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-162*x)*ln(x-81)^8+8*x^2*ln(x-81)^7+(24*x^3-1944*x^2)*ln(x-81)^6+48*x^3*ln(x-81)^5+(96*x^4-7776*x^3
)*ln(x-81)^4+96*x^4*ln(x-81)^3+(160*x^5-12960*x^4)*ln(x-81)^2+64*x^5*ln(x-81)+96*x^6-7776*x^5+x-81)/(x-81),x,m
ethod=_RETURNVERBOSE)

[Out]

ln(x-81)^8*x^2+8*ln(x-81)^6*x^3+24*ln(x-81)^4*x^4+32*ln(x-81)^2*x^5+16*x^6+x

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maxima [B]  time = 0.40, size = 1044, normalized size = 58.00 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-162*x)*log(x-81)^8+8*x^2*log(x-81)^7+(24*x^3-1944*x^2)*log(x-81)^6+48*x^3*log(x-81)^5+(96*x^
4-7776*x^3)*log(x-81)^4+96*x^4*log(x-81)^3+(160*x^5-12960*x^4)*log(x-81)^2+64*x^5*log(x-81)+96*x^6-7776*x^5+x-
81)/(x-81),x, algorithm="maxima")

[Out]

6561*log(x - 81)^8 + 32/25*(25*log(x - 81)^2 - 10*log(x - 81) + 2)*(x - 81)^5 + 16*x^6 + 4251528*log(x - 81)^6
+ 3/4*(32*log(x - 81)^4 - 32*log(x - 81)^3 + 24*log(x - 81)^2 - 12*log(x - 81) + 3)*(x - 81)^4 + 3/4*(32*log(
x - 81)^3 - 24*log(x - 81)^2 + 12*log(x - 81) - 3)*(x - 81)^4 + 1620*(8*log(x - 81)^2 - 4*log(x - 81) + 1)*(x
- 81)^4 - 64/25*x^5 + 8/81*(81*log(x - 81)^6 - 162*log(x - 81)^5 + 270*log(x - 81)^4 - 360*log(x - 81)^3 + 360
*log(x - 81)^2 - 240*log(x - 81) + 80)*(x - 81)^3 + 16/81*(81*log(x - 81)^5 - 135*log(x - 81)^4 + 180*log(x -
81)^3 - 180*log(x - 81)^2 + 120*log(x - 81) - 40)*(x - 81)^3 + 288*(27*log(x - 81)^4 - 36*log(x - 81)^3 + 36*l
og(x - 81)^2 - 24*log(x - 81) + 8)*(x - 81)^3 + 1152*(9*log(x - 81)^3 - 9*log(x - 81)^2 + 6*log(x - 81) - 2)*(
x - 81)^3 + 233280*(9*log(x - 81)^2 - 6*log(x - 81) + 2)*(x - 81)^3 - 2916/5*x^4 + 1033121304*log(x - 81)^4 +
1/2*(2*log(x - 81)^8 - 8*log(x - 81)^7 + 28*log(x - 81)^6 - 84*log(x - 81)^5 + 210*log(x - 81)^4 - 420*log(x -
81)^3 + 630*log(x - 81)^2 - 630*log(x - 81) + 315)*(x - 81)^2 + 1/2*(8*log(x - 81)^7 - 28*log(x - 81)^6 + 84*
log(x - 81)^5 - 210*log(x - 81)^4 + 420*log(x - 81)^3 - 630*log(x - 81)^2 + 630*log(x - 81) - 315)*(x - 81)^2
+ 486*(4*log(x - 81)^6 - 12*log(x - 81)^5 + 30*log(x - 81)^4 - 60*log(x - 81)^3 + 90*log(x - 81)^2 - 90*log(x
- 81) + 45)*(x - 81)^2 + 1458*(4*log(x - 81)^5 - 10*log(x - 81)^4 + 20*log(x - 81)^3 - 30*log(x - 81)^2 + 30*l
og(x - 81) - 15)*(x - 81)^2 + 472392*(2*log(x - 81)^4 - 4*log(x - 81)^3 + 6*log(x - 81)^2 - 6*log(x - 81) + 3)
*(x - 81)^2 + 472392*(4*log(x - 81)^3 - 6*log(x - 81)^2 + 6*log(x - 81) - 3)*(x - 81)^2 + 85030560*(2*log(x -
81)^2 - 2*log(x - 81) + 1)*(x - 81)^2 - 548208/5*x^3 + 162*(log(x - 81)^8 - 8*log(x - 81)^7 + 56*log(x - 81)^6
- 336*log(x - 81)^5 + 1680*log(x - 81)^4 - 6720*log(x - 81)^3 + 20160*log(x - 81)^2 - 40320*log(x - 81) + 403
20)*(x - 81) + 1296*(log(x - 81)^7 - 7*log(x - 81)^6 + 42*log(x - 81)^5 - 210*log(x - 81)^4 + 840*log(x - 81)^
3 - 2520*log(x - 81)^2 + 5040*log(x - 81) - 5040)*(x - 81) + 157464*(log(x - 81)^6 - 6*log(x - 81)^5 + 30*log(
x - 81)^4 - 120*log(x - 81)^3 + 360*log(x - 81)^2 - 720*log(x - 81) + 720)*(x - 81) + 944784*(log(x - 81)^5 -
5*log(x - 81)^4 + 20*log(x - 81)^3 - 60*log(x - 81)^2 + 120*log(x - 81) - 120)*(x - 81) + 51018336*(log(x - 81
)^4 - 4*log(x - 81)^3 + 12*log(x - 81)^2 - 24*log(x - 81) + 24)*(x - 81) + 204073344*(log(x - 81)^3 - 3*log(x
- 81)^2 + 6*log(x - 81) - 6)*(x - 81) + 6887475360*(log(x - 81)^2 - 2*log(x - 81) + 2)*(x - 81) - 109122552/5*
x^2 + 16/5*(4*x^5 + 405*x^4 + 43740*x^3 + 5314410*x^2 + 860934420*x + 69735688020*log(x - 81))*log(x - 81) - 1
11577100832*log(x - 81)^2 - 31452804139/5*x - 2547677135664/5*log(x - 81)

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mupad [B]  time = 1.86, size = 50, normalized size = 2.78 $$\begin{array}{c}16x^6+32x^5{\ln (x-81)}^2+24x^4{\ln (x-81)}^4+8x^3{\ln (x-81)}^6+x^2{\ln (x-81)}^8+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - log(x - 81)^8*(162*x - 2*x^2) + 64*x^5*log(x - 81) - log(x - 81)^6*(1944*x^2 - 24*x^3) - log(x - 81)^
4*(7776*x^3 - 96*x^4) - log(x - 81)^2*(12960*x^4 - 160*x^5) - 7776*x^5 + 96*x^6 + 96*x^4*log(x - 81)^3 + 48*x^
3*log(x - 81)^5 + 8*x^2*log(x - 81)^7 - 81)/(x - 81),x)

[Out]

x + 16*x^6 + 32*x^5*log(x - 81)^2 + 24*x^4*log(x - 81)^4 + 8*x^3*log(x - 81)^6 + x^2*log(x - 81)^8

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sympy [B]  time = 0.20, size = 51, normalized size = 2.83 $$\begin{array}{c}16x^6+32x^5\log {(x-81)}^2+24x^4\log {(x-81)}^4+8x^3\log {(x-81)}^6+x^2\log {(x-81)}^8+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-162*x)*ln(x-81)**8+8*x**2*ln(x-81)**7+(24*x**3-1944*x**2)*ln(x-81)**6+48*x**3*ln(x-81)**5+(
96*x**4-7776*x**3)*ln(x-81)**4+96*x**4*ln(x-81)**3+(160*x**5-12960*x**4)*ln(x-81)**2+64*x**5*ln(x-81)+96*x**6-
7776*x**5+x-81)/(x-81),x)

[Out]

16*x**6 + 32*x**5*log(x - 81)**2 + 24*x**4*log(x - 81)**4 + 8*x**3*log(x - 81)**6 + x**2*log(x - 81)**8 + x

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