3.91.80 \(\int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+(1728 x^2+864 x^3-1728 x^4) \log (x)+(288 x-720 x^2-864 x^3) \log ^2(x)+(16-184 x-144 x^2) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx\)

Optimal. Leaf size=27 \[ \frac {2 \left (2-x \left (-1+x-\left (6+\frac {\log (x)}{x}\right )^4\right )\right )}{2+x} \]

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Rubi [B]  time = 1.43, antiderivative size = 140, normalized size of antiderivative = 5.19, number of steps used = 72, number of rules used = 19, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1594, 27, 6742, 44, 43, 2357, 2304, 2301, 2314, 31, 2317, 2391, 2305, 2302, 30, 2318, 2374, 6589, 2383} \begin {gather*} \frac {\log ^4(x)}{x^3}-\frac {\log ^4(x)}{2 x^2}+\frac {24 \log ^3(x)}{x^2}-2 x-\frac {5192}{x+2}+\frac {\log ^4(x)}{4 x}+\frac {x \log ^4(x)}{8 (x+2)}-\frac {\log ^4(x)}{8}-\frac {12 \log ^3(x)}{x}-\frac {6 x \log ^3(x)}{x+2}+6 \log ^3(x)+\frac {216 \log ^2(x)}{x}+\frac {108 x \log ^2(x)}{x+2}-108 \log ^2(x)-\frac {864 x \log (x)}{x+2}+864 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3456*x^3 + 6912*x^4 - 8*x^5 - 2*x^6 + (1728*x^2 + 864*x^3 - 1728*x^4)*Log[x] + (288*x - 720*x^2 - 864*x^3
)*Log[x]^2 + (16 - 184*x - 144*x^2)*Log[x]^3 + (-12 - 8*x)*Log[x]^4)/(4*x^4 + 4*x^5 + x^6),x]

[Out]

-2*x - 5192/(2 + x) + 864*Log[x] - (864*x*Log[x])/(2 + x) - 108*Log[x]^2 + (216*Log[x]^2)/x + (108*x*Log[x]^2)
/(2 + x) + 6*Log[x]^3 + (24*Log[x]^3)/x^2 - (12*Log[x]^3)/x - (6*x*Log[x]^3)/(2 + x) - Log[x]^4/8 + Log[x]^4/x
^3 - Log[x]^4/(2*x^2) + Log[x]^4/(4*x) + (x*Log[x]^4)/(8*(2 + x))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - 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 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 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 {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{x^4 \left (4+4 x+x^2\right )} \, dx\\ &=\int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{x^4 (2+x)^2} \, dx\\ &=\int \left (\frac {6912}{(2+x)^2}+\frac {3456}{x (2+x)^2}-\frac {8 x}{(2+x)^2}-\frac {2 x^2}{(2+x)^2}-\frac {864 \left (-2-x+2 x^2\right ) \log (x)}{x^2 (2+x)^2}-\frac {144 \left (-2+5 x+6 x^2\right ) \log ^2(x)}{x^3 (2+x)^2}-\frac {8 \left (-2+23 x+18 x^2\right ) \log ^3(x)}{x^4 (2+x)^2}-\frac {4 (3+2 x) \log ^4(x)}{x^4 (2+x)^2}\right ) \, dx\\ &=-\frac {6912}{2+x}-2 \int \frac {x^2}{(2+x)^2} \, dx-4 \int \frac {(3+2 x) \log ^4(x)}{x^4 (2+x)^2} \, dx-8 \int \frac {x}{(2+x)^2} \, dx-8 \int \frac {\left (-2+23 x+18 x^2\right ) \log ^3(x)}{x^4 (2+x)^2} \, dx-144 \int \frac {\left (-2+5 x+6 x^2\right ) \log ^2(x)}{x^3 (2+x)^2} \, dx-864 \int \frac {\left (-2-x+2 x^2\right ) \log (x)}{x^2 (2+x)^2} \, dx+3456 \int \frac {1}{x (2+x)^2} \, dx\\ &=-\frac {6912}{2+x}-2 \int \left (1+\frac {4}{(2+x)^2}-\frac {4}{2+x}\right ) \, dx-4 \int \left (\frac {3 \log ^4(x)}{4 x^4}-\frac {\log ^4(x)}{4 x^3}+\frac {\log ^4(x)}{16 x^2}-\frac {\log ^4(x)}{16 (2+x)^2}\right ) \, dx-8 \int \left (-\frac {2}{(2+x)^2}+\frac {1}{2+x}\right ) \, dx-8 \int \left (-\frac {\log ^3(x)}{2 x^4}+\frac {25 \log ^3(x)}{4 x^3}-\frac {13 \log ^3(x)}{8 x^2}+\frac {\log ^3(x)}{16 x}+\frac {3 \log ^3(x)}{2 (2+x)^2}-\frac {\log ^3(x)}{16 (2+x)}\right ) \, dx-144 \int \left (-\frac {\log ^2(x)}{2 x^3}+\frac {7 \log ^2(x)}{4 x^2}-\frac {\log ^2(x)}{8 x}-\frac {3 \log ^2(x)}{2 (2+x)^2}+\frac {\log ^2(x)}{8 (2+x)}\right ) \, dx-864 \int \left (-\frac {\log (x)}{2 x^2}+\frac {\log (x)}{4 x}+\frac {2 \log (x)}{(2+x)^2}-\frac {\log (x)}{4 (2+x)}\right ) \, dx+3456 \int \left (\frac {1}{4 x}-\frac {1}{2 (2+x)^2}-\frac {1}{4 (2+x)}\right ) \, dx\\ &=-2 x-\frac {5192}{2+x}+864 \log (x)-864 \log (2+x)-\frac {1}{4} \int \frac {\log ^4(x)}{x^2} \, dx+\frac {1}{4} \int \frac {\log ^4(x)}{(2+x)^2} \, dx-\frac {1}{2} \int \frac {\log ^3(x)}{x} \, dx+\frac {1}{2} \int \frac {\log ^3(x)}{2+x} \, dx-3 \int \frac {\log ^4(x)}{x^4} \, dx+4 \int \frac {\log ^3(x)}{x^4} \, dx-12 \int \frac {\log ^3(x)}{(2+x)^2} \, dx+13 \int \frac {\log ^3(x)}{x^2} \, dx+18 \int \frac {\log ^2(x)}{x} \, dx-18 \int \frac {\log ^2(x)}{2+x} \, dx-50 \int \frac {\log ^3(x)}{x^3} \, dx+72 \int \frac {\log ^2(x)}{x^3} \, dx-216 \int \frac {\log (x)}{x} \, dx+216 \int \frac {\log (x)}{2+x} \, dx+216 \int \frac {\log ^2(x)}{(2+x)^2} \, dx-252 \int \frac {\log ^2(x)}{x^2} \, dx+432 \int \frac {\log (x)}{x^2} \, dx-1728 \int \frac {\log (x)}{(2+x)^2} \, dx+\int \frac {\log ^4(x)}{x^3} \, dx\\ &=-\frac {432}{x}-2 x-\frac {5192}{2+x}+864 \log (x)-\frac {432 \log (x)}{x}-\frac {864 x \log (x)}{2+x}+216 \log \left (1+\frac {x}{2}\right ) \log (x)-108 \log ^2(x)-\frac {36 \log ^2(x)}{x^2}+\frac {252 \log ^2(x)}{x}+\frac {108 x \log ^2(x)}{2+x}-18 \log \left (1+\frac {x}{2}\right ) \log ^2(x)-\frac {4 \log ^3(x)}{3 x^3}+\frac {25 \log ^3(x)}{x^2}-\frac {13 \log ^3(x)}{x}-\frac {6 x \log ^3(x)}{2+x}+\frac {1}{2} \log \left (1+\frac {x}{2}\right ) \log ^3(x)+\frac {\log ^4(x)}{x^3}-\frac {\log ^4(x)}{2 x^2}+\frac {\log ^4(x)}{4 x}+\frac {x \log ^4(x)}{8 (2+x)}-864 \log (2+x)-\frac {1}{2} \int \frac {\log ^3(x)}{2+x} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int x^3 \, dx,x,\log (x)\right )-\frac {3}{2} \int \frac {\log \left (1+\frac {x}{2}\right ) \log ^2(x)}{x} \, dx+2 \int \frac {\log ^3(x)}{x^3} \, dx+4 \int \frac {\log ^2(x)}{x^4} \, dx-4 \int \frac {\log ^3(x)}{x^4} \, dx+18 \int \frac {\log ^2(x)}{2+x} \, dx+18 \operatorname {Subst}\left (\int x^2 \, dx,x,\log (x)\right )+36 \int \frac {\log \left (1+\frac {x}{2}\right ) \log (x)}{x} \, dx+39 \int \frac {\log ^2(x)}{x^2} \, dx+72 \int \frac {\log (x)}{x^3} \, dx-75 \int \frac {\log ^2(x)}{x^3} \, dx-216 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx-216 \int \frac {\log (x)}{2+x} \, dx-504 \int \frac {\log (x)}{x^2} \, dx+864 \int \frac {1}{2+x} \, dx-\int \frac {\log ^3(x)}{x^2} \, dx\\ &=-\frac {18}{x^2}+\frac {72}{x}-2 x-\frac {5192}{2+x}+864 \log (x)-\frac {36 \log (x)}{x^2}+\frac {72 \log (x)}{x}-\frac {864 x \log (x)}{2+x}-108 \log ^2(x)-\frac {4 \log ^2(x)}{3 x^3}+\frac {3 \log ^2(x)}{2 x^2}+\frac {213 \log ^2(x)}{x}+\frac {108 x \log ^2(x)}{2+x}+6 \log ^3(x)+\frac {24 \log ^3(x)}{x^2}-\frac {12 \log ^3(x)}{x}-\frac {6 x \log ^3(x)}{2+x}-\frac {\log ^4(x)}{8}+\frac {\log ^4(x)}{x^3}-\frac {\log ^4(x)}{2 x^2}+\frac {\log ^4(x)}{4 x}+\frac {x \log ^4(x)}{8 (2+x)}+216 \text {Li}_2\left (-\frac {x}{2}\right )-36 \log (x) \text {Li}_2\left (-\frac {x}{2}\right )+\frac {3}{2} \log ^2(x) \text {Li}_2\left (-\frac {x}{2}\right )+\frac {3}{2} \int \frac {\log \left (1+\frac {x}{2}\right ) \log ^2(x)}{x} \, dx+\frac {8}{3} \int \frac {\log (x)}{x^4} \, dx+3 \int \frac {\log ^2(x)}{x^3} \, dx-3 \int \frac {\log ^2(x)}{x^2} \, dx-3 \int \frac {\log (x) \text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx-4 \int \frac {\log ^2(x)}{x^4} \, dx-36 \int \frac {\log \left (1+\frac {x}{2}\right ) \log (x)}{x} \, dx+36 \int \frac {\text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx-75 \int \frac {\log (x)}{x^3} \, dx+78 \int \frac {\log (x)}{x^2} \, dx+216 \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {8}{27 x^3}+\frac {3}{4 x^2}-\frac {6}{x}-2 x-\frac {5192}{2+x}+864 \log (x)-\frac {8 \log (x)}{9 x^3}+\frac {3 \log (x)}{2 x^2}-\frac {6 \log (x)}{x}-\frac {864 x \log (x)}{2+x}-108 \log ^2(x)+\frac {216 \log ^2(x)}{x}+\frac {108 x \log ^2(x)}{2+x}+6 \log ^3(x)+\frac {24 \log ^3(x)}{x^2}-\frac {12 \log ^3(x)}{x}-\frac {6 x \log ^3(x)}{2+x}-\frac {\log ^4(x)}{8}+\frac {\log ^4(x)}{x^3}-\frac {\log ^4(x)}{2 x^2}+\frac {\log ^4(x)}{4 x}+\frac {x \log ^4(x)}{8 (2+x)}+36 \text {Li}_3\left (-\frac {x}{2}\right )-3 \log (x) \text {Li}_3\left (-\frac {x}{2}\right )-\frac {8}{3} \int \frac {\log (x)}{x^4} \, dx+3 \int \frac {\log (x)}{x^3} \, dx+3 \int \frac {\log (x) \text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx+3 \int \frac {\text {Li}_3\left (-\frac {x}{2}\right )}{x} \, dx-6 \int \frac {\log (x)}{x^2} \, dx-36 \int \frac {\text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx\\ &=-2 x-\frac {5192}{2+x}+864 \log (x)-\frac {864 x \log (x)}{2+x}-108 \log ^2(x)+\frac {216 \log ^2(x)}{x}+\frac {108 x \log ^2(x)}{2+x}+6 \log ^3(x)+\frac {24 \log ^3(x)}{x^2}-\frac {12 \log ^3(x)}{x}-\frac {6 x \log ^3(x)}{2+x}-\frac {\log ^4(x)}{8}+\frac {\log ^4(x)}{x^3}-\frac {\log ^4(x)}{2 x^2}+\frac {\log ^4(x)}{4 x}+\frac {x \log ^4(x)}{8 (2+x)}+3 \text {Li}_4\left (-\frac {x}{2}\right )-3 \int \frac {\text {Li}_3\left (-\frac {x}{2}\right )}{x} \, dx\\ &=-2 x-\frac {5192}{2+x}+864 \log (x)-\frac {864 x \log (x)}{2+x}-108 \log ^2(x)+\frac {216 \log ^2(x)}{x}+\frac {108 x \log ^2(x)}{2+x}+6 \log ^3(x)+\frac {24 \log ^3(x)}{x^2}-\frac {12 \log ^3(x)}{x}-\frac {6 x \log ^3(x)}{2+x}-\frac {\log ^4(x)}{8}+\frac {\log ^4(x)}{x^3}-\frac {\log ^4(x)}{2 x^2}+\frac {\log ^4(x)}{4 x}+\frac {x \log ^4(x)}{8 (2+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 51, normalized size = 1.89 \begin {gather*} \frac {2 \left (-x^3 \left (2596+2 x+x^2\right )+864 x^3 \log (x)+216 x^2 \log ^2(x)+24 x \log ^3(x)+\log ^4(x)\right )}{x^3 (2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3456*x^3 + 6912*x^4 - 8*x^5 - 2*x^6 + (1728*x^2 + 864*x^3 - 1728*x^4)*Log[x] + (288*x - 720*x^2 - 8
64*x^3)*Log[x]^2 + (16 - 184*x - 144*x^2)*Log[x]^3 + (-12 - 8*x)*Log[x]^4)/(4*x^4 + 4*x^5 + x^6),x]

[Out]

(2*(-(x^3*(2596 + 2*x + x^2)) + 864*x^3*Log[x] + 216*x^2*Log[x]^2 + 24*x*Log[x]^3 + Log[x]^4))/(x^3*(2 + x))

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fricas [B]  time = 0.60, size = 56, normalized size = 2.07 \begin {gather*} -\frac {2 \, {\left (x^{5} + 2 \, x^{4} - 864 \, x^{3} \log \relax (x) - 216 \, x^{2} \log \relax (x)^{2} - 24 \, x \log \relax (x)^{3} - \log \relax (x)^{4} + 2596 \, x^{3}\right )}}{x^{4} + 2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*log(x)^4+(-144*x^2-184*x+16)*log(x)^3+(-864*x^3-720*x^2+288*x)*log(x)^2+(-1728*x^4+864*x^
3+1728*x^2)*log(x)-2*x^6-8*x^5+6912*x^4+3456*x^3)/(x^6+4*x^5+4*x^4),x, algorithm="fricas")

[Out]

-2*(x^5 + 2*x^4 - 864*x^3*log(x) - 216*x^2*log(x)^2 - 24*x*log(x)^3 - log(x)^4 + 2596*x^3)/(x^4 + 2*x^3)

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giac [B]  time = 0.17, size = 82, normalized size = 3.04 \begin {gather*} -\frac {1}{4} \, {\left (\frac {1}{x + 2} - \frac {x^{2} - 2 \, x + 4}{x^{3}}\right )} \log \relax (x)^{4} + 12 \, {\left (\frac {1}{x + 2} - \frac {x - 2}{x^{2}}\right )} \log \relax (x)^{3} - 216 \, {\left (\frac {1}{x + 2} - \frac {1}{x}\right )} \log \relax (x)^{2} - 2 \, x + \frac {1728 \, \log \relax (x)}{x + 2} - \frac {5192}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*log(x)^4+(-144*x^2-184*x+16)*log(x)^3+(-864*x^3-720*x^2+288*x)*log(x)^2+(-1728*x^4+864*x^
3+1728*x^2)*log(x)-2*x^6-8*x^5+6912*x^4+3456*x^3)/(x^6+4*x^5+4*x^4),x, algorithm="giac")

[Out]

-1/4*(1/(x + 2) - (x^2 - 2*x + 4)/x^3)*log(x)^4 + 12*(1/(x + 2) - (x - 2)/x^2)*log(x)^3 - 216*(1/(x + 2) - 1/x
)*log(x)^2 - 2*x + 1728*log(x)/(x + 2) - 5192/(x + 2)

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maple [B]  time = 0.06, size = 68, normalized size = 2.52

method	result	size
risch	\(\frac {2 \ln \relax (x )^{4}}{x^{3} \left (2+x \right )}+\frac {48 \ln \relax (x )^{3}}{x^{2} \left (2+x \right )}+\frac {432 \ln \relax (x )^{2}}{x \left (2+x \right )}+\frac {1728 \ln \relax (x )}{2+x}-\frac {2 \left (x^{2}+2 x +2596\right )}{2+x}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x-12)*ln(x)^4+(-144*x^2-184*x+16)*ln(x)^3+(-864*x^3-720*x^2+288*x)*ln(x)^2+(-1728*x^4+864*x^3+1728*x^
2)*ln(x)-2*x^6-8*x^5+6912*x^4+3456*x^3)/(x^6+4*x^5+4*x^4),x,method=_RETURNVERBOSE)

[Out]

2/x^3/(2+x)*ln(x)^4+48/x^2/(2+x)*ln(x)^3+432/x/(2+x)*ln(x)^2+1728*ln(x)/(2+x)-2*(x^2+2*x+2596)/(2+x)

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maxima [A]  time = 0.39, size = 52, normalized size = 1.93 \begin {gather*} -2 \, x + \frac {2 \, {\left (864 \, x^{3} \log \relax (x) + 216 \, x^{2} \log \relax (x)^{2} + 24 \, x \log \relax (x)^{3} + \log \relax (x)^{4}\right )}}{x^{4} + 2 \, x^{3}} - \frac {5192}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*log(x)^4+(-144*x^2-184*x+16)*log(x)^3+(-864*x^3-720*x^2+288*x)*log(x)^2+(-1728*x^4+864*x^
3+1728*x^2)*log(x)-2*x^6-8*x^5+6912*x^4+3456*x^3)/(x^6+4*x^5+4*x^4),x, algorithm="maxima")

[Out]

-2*x + 2*(864*x^3*log(x) + 216*x^2*log(x)^2 + 24*x*log(x)^3 + log(x)^4)/(x^4 + 2*x^3) - 5192/(x + 2)

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mupad [B]  time = 6.92, size = 48, normalized size = 1.78 \begin {gather*} \frac {2\,\left (-x^5+1296\,x^4+864\,x^3\,\ln \relax (x)+216\,x^2\,{\ln \relax (x)}^2+24\,x\,{\ln \relax (x)}^3+{\ln \relax (x)}^4\right )}{x^3\,\left (x+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^3*(184*x + 144*x^2 - 16) + log(x)^2*(720*x^2 - 288*x + 864*x^3) - log(x)*(1728*x^2 + 864*x^3 - 17
28*x^4) - 3456*x^3 - 6912*x^4 + 8*x^5 + 2*x^6 + log(x)^4*(8*x + 12))/(4*x^4 + 4*x^5 + x^6),x)

[Out]

(2*(24*x*log(x)^3 + 864*x^3*log(x) + log(x)^4 + 216*x^2*log(x)^2 + 1296*x^4 - x^5))/(x^3*(x + 2))

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sympy [B]  time = 0.30, size = 60, normalized size = 2.22 \begin {gather*} - 2 x + \frac {2 \log {\relax (x )}^{4}}{x^{4} + 2 x^{3}} + \frac {48 \log {\relax (x )}^{3}}{x^{3} + 2 x^{2}} + \frac {432 \log {\relax (x )}^{2}}{x^{2} + 2 x} + \frac {1728 \log {\relax (x )}}{x + 2} - \frac {5192}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-12)*ln(x)**4+(-144*x**2-184*x+16)*ln(x)**3+(-864*x**3-720*x**2+288*x)*ln(x)**2+(-1728*x**4+86
4*x**3+1728*x**2)*ln(x)-2*x**6-8*x**5+6912*x**4+3456*x**3)/(x**6+4*x**5+4*x**4),x)

[Out]

-2*x + 2*log(x)**4/(x**4 + 2*x**3) + 48*log(x)**3/(x**3 + 2*x**2) + 432*log(x)**2/(x**2 + 2*x) + 1728*log(x)/(
x + 2) - 5192/(x + 2)

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