3.84.93 \(\int \frac {1179648 x^5+131072 x^6+(393216 x^7+131072 x^8) \log (3 x)+(1572864 x^7+262144 x^8) \log ^2(3 x)+(294912 x^9+98304 x^{10}) \log ^3(3 x)+(737280 x^9+147456 x^{10}) \log ^4(3 x)+(73728 x^{11}+24576 x^{12}) \log ^5(3 x)+(147456 x^{11}+32768 x^{12}) \log ^6(3 x)+(6144 x^{13}+2048 x^{14}) \log ^7(3 x)+(10752 x^{13}+2560 x^{14}) \log ^8(3 x)}{243+405 x+270 x^2+90 x^3+15 x^4+x^5} \, dx\)

Optimal. Leaf size=26 \[ \frac {256 x^6 \left (4+x^2 \log ^2(3 x)\right )^4}{(-3-x)^4} \]

________________________________________________________________________________________

Rubi [B]  time = 21.96, antiderivative size = 555, normalized size of antiderivative = 21.35, number of steps used = 888, number of rules used = 27, integrand size = 179, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.151, Rules used = {6688, 12, 6742, 74, 43, 2351, 2295, 2304, 2319, 44, 2314, 31, 2317, 2391, 2357, 2296, 2305, 2347, 2344, 2301, 2318, 2353, 2302, 30, 2374, 6589, 2383} \begin {gather*} 256 x^{10} \log ^8(3 x)-3072 x^9 \log ^8(3 x)+23040 x^8 \log ^8(3 x)+4096 x^8 \log ^6(3 x)-138240 x^7 \log ^8(3 x)-49152 x^7 \log ^6(3 x)+\frac {65536 x^6}{(x+3)^4}+725760 x^6 \log ^8(3 x)+368640 x^6 \log ^6(3 x)+24576 x^6 \log ^4(3 x)-3483648 x^5 \log ^8(3 x)-2211840 x^5 \log ^6(3 x)-294912 x^5 \log ^4(3 x)[phone] x^4 \log ^8(3 x)[phone] x^4 \log ^6(3 x)+2211840 x^4 \log ^4(3 x)+65536 x^4 \log ^2(3 x)-67184640 x^3 \log ^8(3 x)-55738368 x^3 \log ^6(3 x)-13271040 x^3 \log ^4(3 x)-786432 x^3 \log ^2(3 x)[phone] x^2 \log ^8(3 x)[phone] x^2 \log ^6(3 x)[phone] x^2 \log ^4(3 x)+5898240 x^2 \log ^2(3 x)+\frac {5502422016 x \log ^8(3 x)}{x+3}-1108546560 x \log ^8(3 x)+\frac {12380449536 \log ^8(3 x)}{(x+3)^2}-\frac {5714053632 \log ^8(3 x)}{(x+3)^3}+\frac {1224440064 \log ^8(3 x)}{(x+3)^4}-1179090432 \log ^8(3 x)+\frac {5912248320 x \log ^6(3 x)}{x+3}-1074954240 x \log ^6(3 x)+\frac {15963070464 \log ^6(3 x)}{(x+3)^2}-\frac {8707129344 \log ^6(3 x)}{(x+3)^3}+\frac {2176782336 \log ^6(3 x)}{(x+3)^4}-1478062080 \log ^6(3 x)+\frac {2149908480 x \log ^4(3 x)}{x+3}-334430208 x \log ^4(3 x)+\frac {7255941120 \log ^4(3 x)}{(x+3)^2}-\frac {4837294080 \log ^4(3 x)}{(x+3)^3}+\frac {1451188224 \log ^4(3 x)}{(x+3)^4}-644972544 \log ^4(3 x)+\frac {297271296 x \log ^2(3 x)}{x+3}-35389440 x \log ^2(3 x)+\frac {1337720832 \log ^2(3 x)}{(x+3)^2}-\frac {1146617856 \log ^2(3 x)}{(x+3)^3}+\frac {429981696 \log ^2(3 x)}{(x+3)^4}-111476736 \log ^2(3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1179648*x^5 + 131072*x^6 + (393216*x^7 + 131072*x^8)*Log[3*x] + (1572864*x^7 + 262144*x^8)*Log[3*x]^2 + (
294912*x^9 + 98304*x^10)*Log[3*x]^3 + (737280*x^9 + 147456*x^10)*Log[3*x]^4 + (73728*x^11 + 24576*x^12)*Log[3*
x]^5 + (147456*x^11 + 32768*x^12)*Log[3*x]^6 + (6144*x^13 + 2048*x^14)*Log[3*x]^7 + (10752*x^13 + 2560*x^14)*L
og[3*x]^8)/(243 + 405*x + 270*x^2 + 90*x^3 + 15*x^4 + x^5),x]

[Out]

(65536*x^6)/(3 + x)^4 - 111476736*Log[3*x]^2 - 35389440*x*Log[3*x]^2 + 5898240*x^2*Log[3*x]^2 - 786432*x^3*Log
[3*x]^2 + 65536*x^4*Log[3*x]^2 + (429981696*Log[3*x]^2)/(3 + x)^4 - (1146617856*Log[3*x]^2)/(3 + x)^3 + (13377
20832*Log[3*x]^2)/(3 + x)^2 + (297271296*x*Log[3*x]^2)/(3 + x) - 644972544*Log[3*x]^4 - 334430208*x*Log[3*x]^4
+ 69672960*x^2*Log[3*x]^4 - 13271040*x^3*Log[3*x]^4 + 2211840*x^4*Log[3*x]^4 - 294912*x^5*Log[3*x]^4 + 24576*
x^6*Log[3*x]^4 + (1451188224*Log[3*x]^4)/(3 + x)^4 - (4837294080*Log[3*x]^4)/(3 + x)^3 + (7255941120*Log[3*x]^
4)/(3 + x)^2 + (2149908480*x*Log[3*x]^4)/(3 + x) - 1478062080*Log[3*x]^6 - 1074954240*x*Log[3*x]^6 + 250822656
*x^2*Log[3*x]^6 - 55738368*x^3*Log[3*x]^6 + 11612160*x^4*Log[3*x]^6 - 2211840*x^5*Log[3*x]^6 + 368640*x^6*Log[
3*x]^6 - 49152*x^7*Log[3*x]^6 + 4096*x^8*Log[3*x]^6 + (2176782336*Log[3*x]^6)/(3 + x)^4 - (8707129344*Log[3*x]
^6)/(3 + x)^3 + (15963070464*Log[3*x]^6)/(3 + x)^2 + (5912248320*x*Log[3*x]^6)/(3 + x) - 1179090432*Log[3*x]^8
- 1108546560*x*Log[3*x]^8 + 277136640*x^2*Log[3*x]^8 - 67184640*x^3*Log[3*x]^8 + 15676416*x^4*Log[3*x]^8 - 34
83648*x^5*Log[3*x]^8 + 725760*x^6*Log[3*x]^8 - 138240*x^7*Log[3*x]^8 + 23040*x^8*Log[3*x]^8 - 3072*x^9*Log[3*x
]^8 + 256*x^10*Log[3*x]^8 + (1224440064*Log[3*x]^8)/(3 + x)^4 - (5714053632*Log[3*x]^8)/(3 + x)^3 + (123804495
36*Log[3*x]^8)/(3 + x)^2 + (5502422016*x*Log[3*x]^8)/(3 + x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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 2314

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

Rule 2317

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

Rule 2318

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

Rule 2319

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

Rule 2344

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

Rule 2347

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

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2353

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

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6589

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

Rule 6688

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

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 \frac {512 x^5 \left (4+x^2 \log ^2(3 x)\right )^3 \left (4 (9+x)+4 x^2 (3+x) \log (3 x)+x^2 (21+5 x) \log ^2(3 x)\right )}{(3+x)^5} \, dx\\ &=512 \int \frac {x^5 \left (4+x^2 \log ^2(3 x)\right )^3 \left (4 (9+x)+4 x^2 (3+x) \log (3 x)+x^2 (21+5 x) \log ^2(3 x)\right )}{(3+x)^5} \, dx\\ &=512 \int \left (\frac {256 x^5 (9+x)}{(3+x)^5}+\frac {256 x^7 \log (3 x)}{(3+x)^4}+\frac {512 x^7 (6+x) \log ^2(3 x)}{(3+x)^5}+\frac {192 x^9 \log ^3(3 x)}{(3+x)^4}+\frac {288 x^9 (5+x) \log ^4(3 x)}{(3+x)^5}+\frac {48 x^{11} \log ^5(3 x)}{(3+x)^4}+\frac {32 x^{11} (9+2 x) \log ^6(3 x)}{(3+x)^5}+\frac {4 x^{13} \log ^7(3 x)}{(3+x)^4}+\frac {x^{13} (21+5 x) \log ^8(3 x)}{(3+x)^5}\right ) \, dx\\ &=512 \int \frac {x^{13} (21+5 x) \log ^8(3 x)}{(3+x)^5} \, dx+2048 \int \frac {x^{13} \log ^7(3 x)}{(3+x)^4} \, dx+16384 \int \frac {x^{11} (9+2 x) \log ^6(3 x)}{(3+x)^5} \, dx+24576 \int \frac {x^{11} \log ^5(3 x)}{(3+x)^4} \, dx+98304 \int \frac {x^9 \log ^3(3 x)}{(3+x)^4} \, dx+131072 \int \frac {x^5 (9+x)}{(3+x)^5} \, dx+131072 \int \frac {x^7 \log (3 x)}{(3+x)^4} \, dx+147456 \int \frac {x^9 (5+x) \log ^4(3 x)}{(3+x)^5} \, dx+262144 \int \frac {x^7 (6+x) \log ^2(3 x)}{(3+x)^5} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 5.14, size = 76, normalized size = 2.92 \begin {gather*} \frac {256 \left (256 \left (-7290-9720 x-4860 x^2-1080 x^3-90 x^4+x^6\right )+256 x^8 \log ^2(3 x)+96 x^{10} \log ^4(3 x)+16 x^{12} \log ^6(3 x)+x^{14} \log ^8(3 x)\right )}{(3+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1179648*x^5 + 131072*x^6 + (393216*x^7 + 131072*x^8)*Log[3*x] + (1572864*x^7 + 262144*x^8)*Log[3*x]
^2 + (294912*x^9 + 98304*x^10)*Log[3*x]^3 + (737280*x^9 + 147456*x^10)*Log[3*x]^4 + (73728*x^11 + 24576*x^12)*
Log[3*x]^5 + (147456*x^11 + 32768*x^12)*Log[3*x]^6 + (6144*x^13 + 2048*x^14)*Log[3*x]^7 + (10752*x^13 + 2560*x
^14)*Log[3*x]^8)/(243 + 405*x + 270*x^2 + 90*x^3 + 15*x^4 + x^5),x]

[Out]

(256*(256*(-7290 - 9720*x - 4860*x^2 - 1080*x^3 - 90*x^4 + x^6) + 256*x^8*Log[3*x]^2 + 96*x^10*Log[3*x]^4 + 16
*x^12*Log[3*x]^6 + x^14*Log[3*x]^8))/(3 + x)^4

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fricas [B]  time = 0.64, size = 90, normalized size = 3.46 \begin {gather*} \frac {256 \, {\left (x^{14} \log \left (3 \, x\right )^{8} + 16 \, x^{12} \log \left (3 \, x\right )^{6} + 96 \, x^{10} \log \left (3 \, x\right )^{4} + 256 \, x^{8} \log \left (3 \, x\right )^{2} + 256 \, x^{6} - 23040 \, x^{4} - 276480 \, x^{3} - 1244160 \, x^{2} - 2488320 \, x - 1866240\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2560*x^14+10752*x^13)*log(3*x)^8+(2048*x^14+6144*x^13)*log(3*x)^7+(32768*x^12+147456*x^11)*log(3*x
)^6+(24576*x^12+73728*x^11)*log(3*x)^5+(147456*x^10+737280*x^9)*log(3*x)^4+(98304*x^10+294912*x^9)*log(3*x)^3+
(262144*x^8+1572864*x^7)*log(3*x)^2+(131072*x^8+393216*x^7)*log(3*x)+131072*x^6+1179648*x^5)/(x^5+15*x^4+90*x^
3+270*x^2+405*x+243),x, algorithm="fricas")

[Out]

256*(x^14*log(3*x)^8 + 16*x^12*log(3*x)^6 + 96*x^10*log(3*x)^4 + 256*x^8*log(3*x)^2 + 256*x^6 - 23040*x^4 - 27
6480*x^3 - 1244160*x^2 - 2488320*x - 1866240)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81)

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giac [B]  time = 0.21, size = 358, normalized size = 13.77 \begin {gather*} 256 \, {\left (x^{10} - 12 \, x^{9} + 90 \, x^{8} - 540 \, x^{7} + 2835 \, x^{6} - 13608 \, x^{5} + 61236 \, x^{4} - 262440 \, x^{3} + 1082565 \, x^{2} - 4330260 \, x - \frac {177147 \, {\left (364 \, x^{3} + 3003 \, x^{2} + 8316 \, x + 7722\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 16888014\right )} \log \left (3 \, x\right )^{8} + 4096 \, {\left (x^{8} - 12 \, x^{7} + 90 \, x^{6} - 540 \, x^{5} + 2835 \, x^{4} - 13608 \, x^{3} + 61236 \, x^{2} - 262440 \, x - \frac {19683 \, {\left (220 \, x^{3} + 1782 \, x^{2} + 4860 \, x + 4455\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 1082565\right )} \log \left (3 \, x\right )^{6} + 24576 \, {\left (x^{6} - 12 \, x^{5} + 90 \, x^{4} - 540 \, x^{3} + 2835 \, x^{2} - 13608 \, x - \frac {6561 \, {\left (40 \, x^{3} + 315 \, x^{2} + 840 \, x + 756\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 61236\right )} \log \left (3 \, x\right )^{4} + 65536 \, {\left (x^{4} - 12 \, x^{3} + 90 \, x^{2} - 540 \, x - \frac {243 \, {\left (56 \, x^{3} + 420 \, x^{2} + 1080 \, x + 945\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + 2835\right )} \log \left (3 \, x\right )^{2} + 65536 \, x^{2} - 786432 \, x - \frac {1769472 \, {\left (20 \, x^{3} + 135 \, x^{2} + 324 \, x + 270\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2560*x^14+10752*x^13)*log(3*x)^8+(2048*x^14+6144*x^13)*log(3*x)^7+(32768*x^12+147456*x^11)*log(3*x
)^6+(24576*x^12+73728*x^11)*log(3*x)^5+(147456*x^10+737280*x^9)*log(3*x)^4+(98304*x^10+294912*x^9)*log(3*x)^3+
(262144*x^8+1572864*x^7)*log(3*x)^2+(131072*x^8+393216*x^7)*log(3*x)+131072*x^6+1179648*x^5)/(x^5+15*x^4+90*x^
3+270*x^2+405*x+243),x, algorithm="giac")

[Out]

256*(x^10 - 12*x^9 + 90*x^8 - 540*x^7 + 2835*x^6 - 13608*x^5 + 61236*x^4 - 262440*x^3 + 1082565*x^2 - 4330260*
x - 177147*(364*x^3 + 3003*x^2 + 8316*x + 7722)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81) + 16888014)*log(3*x)^8 +
4096*(x^8 - 12*x^7 + 90*x^6 - 540*x^5 + 2835*x^4 - 13608*x^3 + 61236*x^2 - 262440*x - 19683*(220*x^3 + 1782*x^
2 + 4860*x + 4455)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81) + 1082565)*log(3*x)^6 + 24576*(x^6 - 12*x^5 + 90*x^4 -
540*x^3 + 2835*x^2 - 13608*x - 6561*(40*x^3 + 315*x^2 + 840*x + 756)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81) + 6
1236)*log(3*x)^4 + 65536*(x^4 - 12*x^3 + 90*x^2 - 540*x - 243*(56*x^3 + 420*x^2 + 1080*x + 945)/(x^4 + 12*x^3
+ 54*x^2 + 108*x + 81) + 2835)*log(3*x)^2 + 65536*x^2 - 786432*x - 1769472*(20*x^3 + 135*x^2 + 324*x + 270)/(x
^4 + 12*x^3 + 54*x^2 + 108*x + 81)

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maple [B]  time = 0.52, size = 171, normalized size = 6.58

method	result	size
risch	\(\frac {256 x^{14} \ln \left (3 x \right )^{8}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {4096 x^{12} \ln \left (3 x \right )^{6}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {24576 x^{10} \ln \left (3 x \right )^{4}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {65536 x^{8} \ln \left (3 x \right )^{2}}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}+\frac {65536 x^{6}-5898240 x^{4}-70778880 x^{3}-318504960 x^{2}-637009920 x -477757440}{x^{4}+12 x^{3}+54 x^{2}+108 x +81}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2560*x^14+10752*x^13)*ln(3*x)^8+(2048*x^14+6144*x^13)*ln(3*x)^7+(32768*x^12+147456*x^11)*ln(3*x)^6+(2457
6*x^12+73728*x^11)*ln(3*x)^5+(147456*x^10+737280*x^9)*ln(3*x)^4+(98304*x^10+294912*x^9)*ln(3*x)^3+(262144*x^8+
1572864*x^7)*ln(3*x)^2+(131072*x^8+393216*x^7)*ln(3*x)+131072*x^6+1179648*x^5)/(x^5+15*x^4+90*x^3+270*x^2+405*
x+243),x,method=_RETURNVERBOSE)

[Out]

256*x^14/(x^4+12*x^3+54*x^2+108*x+81)*ln(3*x)^8+4096*x^12/(x^4+12*x^3+54*x^2+108*x+81)*ln(3*x)^6+24576*x^10/(x
^4+12*x^3+54*x^2+108*x+81)*ln(3*x)^4+65536*x^8/(x^4+12*x^3+54*x^2+108*x+81)*ln(3*x)^2+65536*(x^6-90*x^4-1080*x
^3-4860*x^2-9720*x-7290)/(x^4+12*x^3+54*x^2+108*x+81)

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maxima [B]  time = 0.53, size = 485, normalized size = 18.65 \begin {gather*} 65536 \, x^{2} - 786432 \, x + \frac {256 \, {\left (x^{14} \log \relax (3)^{8} + 8 \, x^{14} \log \relax (3) \log \relax (x)^{7} + x^{14} \log \relax (x)^{8} + 16 \, x^{12} \log \relax (3)^{6} + 96 \, x^{10} \log \relax (3)^{4} + 256 \, x^{8} \log \relax (3)^{2} - 1367929134 \, \log \relax (3)^{8} + 4 \, {\left (7 \, x^{14} \log \relax (3)^{2} + 4 \, x^{12}\right )} \log \relax (x)^{6} - 1403004240 \, \log \relax (3)^{6} + 8 \, {\left (7 \, x^{14} \log \relax (3)^{3} + 12 \, x^{12} \log \relax (3)\right )} \log \relax (x)^{5} - 162 \, {\left (104247 \, \log \relax (3)^{8} + 106920 \, \log \relax (3)^{6} + 36288 \, \log \relax (3)^{4} + 4480 \, \log \relax (3)^{2}\right )} x^{4} + 2 \, {\left (35 \, x^{14} \log \relax (3)^{4} + 120 \, x^{12} \log \relax (3)^{2} + 48 \, x^{10}\right )} \log \relax (x)^{4} - 1944 \, {\left (104247 \, \log \relax (3)^{8} + 106920 \, \log \relax (3)^{6} + 36288 \, \log \relax (3)^{4} + 4480 \, \log \relax (3)^{2}\right )} x^{3} - 476171136 \, \log \relax (3)^{4} + 8 \, {\left (7 \, x^{14} \log \relax (3)^{5} + 40 \, x^{12} \log \relax (3)^{3} + 48 \, x^{10} \log \relax (3)\right )} \log \relax (x)^{3} - 8748 \, {\left (104247 \, \log \relax (3)^{8} + 106920 \, \log \relax (3)^{6} + 36288 \, \log \relax (3)^{4} + 4480 \, \log \relax (3)^{2}\right )} x^{2} + 4 \, {\left (7 \, x^{14} \log \relax (3)^{6} + 60 \, x^{12} \log \relax (3)^{4} + 144 \, x^{10} \log \relax (3)^{2} + 64 \, x^{8}\right )} \log \relax (x)^{2} - 17496 \, {\left (104247 \, \log \relax (3)^{8} + 106920 \, \log \relax (3)^{6} + 36288 \, \log \relax (3)^{4} + 4480 \, \log \relax (3)^{2}\right )} x - 58786560 \, \log \relax (3)^{2} + 8 \, {\left (x^{14} \log \relax (3)^{7} + 12 \, x^{12} \log \relax (3)^{5} + 48 \, x^{10} \log \relax (3)^{3} + 64 \, x^{8} \log \relax (3)\right )} \log \relax (x)\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} + \frac {884736 \, {\left (80 \, x^{3} + 630 \, x^{2} + 1692 \, x + 1539\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} - \frac {2654208 \, {\left (40 \, x^{3} + 300 \, x^{2} + 780 \, x + 693\right )}}{x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2560*x^14+10752*x^13)*log(3*x)^8+(2048*x^14+6144*x^13)*log(3*x)^7+(32768*x^12+147456*x^11)*log(3*x
)^6+(24576*x^12+73728*x^11)*log(3*x)^5+(147456*x^10+737280*x^9)*log(3*x)^4+(98304*x^10+294912*x^9)*log(3*x)^3+
(262144*x^8+1572864*x^7)*log(3*x)^2+(131072*x^8+393216*x^7)*log(3*x)+131072*x^6+1179648*x^5)/(x^5+15*x^4+90*x^
3+270*x^2+405*x+243),x, algorithm="maxima")

[Out]

65536*x^2 - 786432*x + 256*(x^14*log(3)^8 + 8*x^14*log(3)*log(x)^7 + x^14*log(x)^8 + 16*x^12*log(3)^6 + 96*x^1
0*log(3)^4 + 256*x^8*log(3)^2 - 1367929134*log(3)^8 + 4*(7*x^14*log(3)^2 + 4*x^12)*log(x)^6 - 1403004240*log(3
)^6 + 8*(7*x^14*log(3)^3 + 12*x^12*log(3))*log(x)^5 - 162*(104247*log(3)^8 + 106920*log(3)^6 + 36288*log(3)^4
+ 4480*log(3)^2)*x^4 + 2*(35*x^14*log(3)^4 + 120*x^12*log(3)^2 + 48*x^10)*log(x)^4 - 1944*(104247*log(3)^8 + 1
06920*log(3)^6 + 36288*log(3)^4 + 4480*log(3)^2)*x^3 - 476171136*log(3)^4 + 8*(7*x^14*log(3)^5 + 40*x^12*log(3
)^3 + 48*x^10*log(3))*log(x)^3 - 8748*(104247*log(3)^8 + 106920*log(3)^6 + 36288*log(3)^4 + 4480*log(3)^2)*x^2
+ 4*(7*x^14*log(3)^6 + 60*x^12*log(3)^4 + 144*x^10*log(3)^2 + 64*x^8)*log(x)^2 - 17496*(104247*log(3)^8 + 106
920*log(3)^6 + 36288*log(3)^4 + 4480*log(3)^2)*x - 58786560*log(3)^2 + 8*(x^14*log(3)^7 + 12*x^12*log(3)^5 + 4
8*x^10*log(3)^3 + 64*x^8*log(3))*log(x))/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81) + 884736*(80*x^3 + 630*x^2 + 169
2*x + 1539)/(x^4 + 12*x^3 + 54*x^2 + 108*x + 81) - 2654208*(40*x^3 + 300*x^2 + 780*x + 693)/(x^4 + 12*x^3 + 54
*x^2 + 108*x + 81)

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mupad [B]  time = 5.57, size = 170, normalized size = 6.54 \begin {gather*} 65536\,x^2-\frac {35389440\,x^3+238878720\,x^2+573308928\,x+477757440}{x^4+12\,x^3+54\,x^2+108\,x+81}-786432\,x+\frac {65536\,x^8\,{\ln \left (3\,x\right )}^2}{x^4+12\,x^3+54\,x^2+108\,x+81}+\frac {24576\,x^{10}\,{\ln \left (3\,x\right )}^4}{x^4+12\,x^3+54\,x^2+108\,x+81}+\frac {4096\,x^{12}\,{\ln \left (3\,x\right )}^6}{x^4+12\,x^3+54\,x^2+108\,x+81}+\frac {256\,x^{14}\,{\ln \left (3\,x\right )}^8}{x^4+12\,x^3+54\,x^2+108\,x+81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3*x)*(393216*x^7 + 131072*x^8) + log(3*x)^7*(6144*x^13 + 2048*x^14) + log(3*x)^8*(10752*x^13 + 2560*x
^14) + log(3*x)^5*(73728*x^11 + 24576*x^12) + log(3*x)^6*(147456*x^11 + 32768*x^12) + log(3*x)^3*(294912*x^9 +
98304*x^10) + log(3*x)^4*(737280*x^9 + 147456*x^10) + log(3*x)^2*(1572864*x^7 + 262144*x^8) + 1179648*x^5 + 1
31072*x^6)/(405*x + 270*x^2 + 90*x^3 + 15*x^4 + x^5 + 243),x)

[Out]

65536*x^2 - (573308928*x + 238878720*x^2 + 35389440*x^3 + 477757440)/(108*x + 54*x^2 + 12*x^3 + x^4 + 81) - 78
6432*x + (65536*x^8*log(3*x)^2)/(108*x + 54*x^2 + 12*x^3 + x^4 + 81) + (24576*x^10*log(3*x)^4)/(108*x + 54*x^2
+ 12*x^3 + x^4 + 81) + (4096*x^12*log(3*x)^6)/(108*x + 54*x^2 + 12*x^3 + x^4 + 81) + (256*x^14*log(3*x)^8)/(1
08*x + 54*x^2 + 12*x^3 + x^4 + 81)

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sympy [B]  time = 0.98, size = 165, normalized size = 6.35 \begin {gather*} \frac {256 x^{14} \log {\left (3 x \right )}^{8}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + \frac {4096 x^{12} \log {\left (3 x \right )}^{6}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + \frac {24576 x^{10} \log {\left (3 x \right )}^{4}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + \frac {65536 x^{8} \log {\left (3 x \right )}^{2}}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} + 65536 x^{2} - 786432 x + \frac {- 35389440 x^{3} - 238878720 x^{2} - 573308928 x - 477757440}{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2560*x**14+10752*x**13)*ln(3*x)**8+(2048*x**14+6144*x**13)*ln(3*x)**7+(32768*x**12+147456*x**11)*l
n(3*x)**6+(24576*x**12+73728*x**11)*ln(3*x)**5+(147456*x**10+737280*x**9)*ln(3*x)**4+(98304*x**10+294912*x**9)
*ln(3*x)**3+(262144*x**8+1572864*x**7)*ln(3*x)**2+(131072*x**8+393216*x**7)*ln(3*x)+131072*x**6+1179648*x**5)/
(x**5+15*x**4+90*x**3+270*x**2+405*x+243),x)

[Out]

256*x**14*log(3*x)**8/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81) + 4096*x**12*log(3*x)**6/(x**4 + 12*x**3 + 54*x*
*2 + 108*x + 81) + 24576*x**10*log(3*x)**4/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81) + 65536*x**8*log(3*x)**2/(x
**4 + 12*x**3 + 54*x**2 + 108*x + 81) + 65536*x**2 - 786432*x + (-35389440*x**3 - 238878720*x**2 - 573308928*x
- 477757440)/(x**4 + 12*x**3 + 54*x**2 + 108*x + 81)

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