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

Optimal. Leaf size=18 \[ x+x^2 \left (2 x+\log ^2(-81+x)\right )^4 \]

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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 {gather*} 16 x^6+32 x^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] \log ^4(x-81) \end {gather*}

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

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

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 {gather*} x^{2} \log \left (x - 81\right )^{8} + 8 \, x^{3} \log \left (x - 81\right )^{6} + 24 \, x^{4} \log \left (x - 81\right )^{4} + 32 \, x^{5} \log \left (x - 81\right )^{2} + 16 \, x^{6} + x \end {gather*}

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 {gather*} x^{2} \log \left (x - 81\right )^{8} + 8 \, x^{3} \log \left (x - 81\right )^{6} + 24 \, x^{4} \log \left (x - 81\right )^{4} + 32 \, x^{5} \log \left (x - 81\right )^{2} + 16 \, x^{6} + x \end {gather*}

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	\(\ln \left (x -81\right )^{8} x^{2}+8 \ln \left (x -81\right )^{6} x^{3}+24 \ln \left (x -81\right )^{4} x^{4}+32 \ln \left (x -81\right )^{2} x^{5}+16 x^{6}+x\)	\(51\)
derivativedivides	\(-27113235502257+334731302497 x [phone] \ln \left (x -81\right )^{2}[phone] \left (x -81\right )^{2}+\ln \left (x -81\right )^{8} \left (x -81\right )^{2}+162 \ln \left (x -81\right )^{8} \left (x -81\right )+8 \ln \left (x -81\right )^{6} \left (x -81\right )^{3}+1944 \ln \left (x -81\right )^{6} \left (x -81\right )^{2}+24 \ln \left (x -81\right )^{4} \left (x -81\right )^{4}+157464 \ln \left (x -81\right )^{6} \left (x -81\right )+7776 \ln \left (x -81\right )^{4} \left (x -81\right )^{3}+32 \ln \left (x -81\right )^{2} \left (x -81\right )^{5}+944784 \ln \left (x -81\right )^{4} \left (x -81\right )^{2}+12960 \ln \left (x -81\right )^{2} \left (x -81\right )^{4}[phone] \ln \left (x -81\right )^{4} \left (x -81\right )+2099520 \ln \left (x -81\right )^{2} \left (x -81\right )^{3}[phone] \ln \left (x -81\right )^{2} \left (x -81\right )^{2}[phone] \ln \left (x -81\right )^{2} \left (x -81\right )+6561 \ln \left (x -81\right )^{8}+4251528 \ln \left (x -81\right )^{6}[phone] \ln \left (x -81\right )^{4}+16 \left (x -81\right )^{6}+7776 \left (x -81\right )^{5}+1574640 \left (x -81\right )^{4}[phone] \left (x -81\right )^{3}\)	\(246\)
default	\(-27113235502257+334731302497 x [phone] \ln \left (x -81\right )^{2}[phone] \left (x -81\right )^{2}+\ln \left (x -81\right )^{8} \left (x -81\right )^{2}+162 \ln \left (x -81\right )^{8} \left (x -81\right )+8 \ln \left (x -81\right )^{6} \left (x -81\right )^{3}+1944 \ln \left (x -81\right )^{6} \left (x -81\right )^{2}+24 \ln \left (x -81\right )^{4} \left (x -81\right )^{4}+157464 \ln \left (x -81\right )^{6} \left (x -81\right )+7776 \ln \left (x -81\right )^{4} \left (x -81\right )^{3}+32 \ln \left (x -81\right )^{2} \left (x -81\right )^{5}+944784 \ln \left (x -81\right )^{4} \left (x -81\right )^{2}+12960 \ln \left (x -81\right )^{2} \left (x -81\right )^{4}[phone] \ln \left (x -81\right )^{4} \left (x -81\right )+2099520 \ln \left (x -81\right )^{2} \left (x -81\right )^{3}[phone] \ln \left (x -81\right )^{2} \left (x -81\right )^{2}[phone] \ln \left (x -81\right )^{2} \left (x -81\right )+6561 \ln \left (x -81\right )^{8}+4251528 \ln \left (x -81\right )^{6}[phone] \ln \left (x -81\right )^{4}+16 \left (x -81\right )^{6}+7776 \left (x -81\right )^{5}+1574640 \left (x -81\right )^{4}[phone] \left (x -81\right )^{3}\)	\(246\)

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 {gather*} 16\,x^6+32\,x^5\,{\ln \left (x-81\right )}^2+24\,x^4\,{\ln \left (x-81\right )}^4+8\,x^3\,{\ln \left (x-81\right )}^6+x^2\,{\ln \left (x-81\right )}^8+x \end {gather*}

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 {gather*} 16 x^{6} + 32 x^{5} \log {\left (x - 81 \right )}^{2} + 24 x^{4} \log {\left (x - 81 \right )}^{4} + 8 x^{3} \log {\left (x - 81 \right )}^{6} + x^{2} \log {\left (x - 81 \right )}^{8} + x \end {gather*}

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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