∫ (5x+12x2−27x3+12x4+5x5) log(x)+(1+5x+6x2−9x3+3x4+x5) log(1+5x+6x2−9x3+3x4+x5)__________________________________________________________________

3.29.63 \(\int \frac {(5 x+12 x^2-27 x^3+12 x^4+5 x^5) \log (x)+(1+5 x+6 x^2-9 x^3+3 x^4+x^5) \log (1+5 x+6 x^2-9 x^3+3 x^4+x^5)}{x+5 x^2+6 x^3-9 x^4+3 x^5+x^6} \, dx\)

Optimal. Leaf size=21 \[ \log (x) \log \left (1+(5+x) \left (x+\left (-x+x^2\right )^2\right )\right ) \]

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Rubi [A] time = 0.77, antiderivative size = 27, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 3, integrand size = 104, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6688, 2357, 2524} \begin {gather*} \log (x) \log \left (x^5+3 x^4-9 x^3+6 x^2+5 x+1\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[((5*x + 12*x^2 - 27*x^3 + 12*x^4 + 5*x^5)*Log[x] + (1 + 5*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5)*Log[1 + 5*x + 6

*x^2 - 9*x^3 + 3*x^4 + x^5])/(x + 5*x^2 + 6*x^3 - 9*x^4 + 3*x^5 + x^6),x]

[Out]

Log[x]*Log[1 + 5*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5]

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 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b

*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x

] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\left (5+12 x-27 x^2+12 x^3+5 x^4\right ) \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}+\frac {\log \left (1+5 x+6 x^2-9 x^3+3 x^4+x^5\right )}{x}\right ) \, dx\\ &=\int \frac {\left (5+12 x-27 x^2+12 x^3+5 x^4\right ) \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5} \, dx+\int \frac {\log \left (1+5 x+6 x^2-9 x^3+3 x^4+x^5\right )}{x} \, dx\\ &=\log (x) \log \left (1+5 x+6 x^2-9 x^3+3 x^4+x^5\right )-\int \frac {\left (5+12 x-27 x^2+12 x^3+5 x^4\right ) \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5} \, dx+\int \left (\frac {5 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}+\frac {12 x \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}-\frac {27 x^2 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}+\frac {12 x^3 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}+\frac {5 x^4 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}\right ) \, dx\\ &=\log (x) \log \left (1+5 x+6 x^2-9 x^3+3 x^4+x^5\right )+5 \int \frac {\log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5} \, dx+5 \int \frac {x^4 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5} \, dx+12 \int \frac {x \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5} \, dx+12 \int \frac {x^3 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5} \, dx-27 \int \frac {x^2 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5} \, dx-\int \left (\frac {5 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}+\frac {12 x \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}-\frac {27 x^2 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}+\frac {12 x^3 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}+\frac {5 x^4 \log (x)}{1+5 x+6 x^2-9 x^3+3 x^4+x^5}\right ) \, dx\\ &=\log (x) \log \left (1+5 x+6 x^2-9 x^3+3 x^4+x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [C] time = 2.31, size = 3382, normalized size = 161.05 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((5*x + 12*x^2 - 27*x^3 + 12*x^4 + 5*x^5)*Log[x] + (1 + 5*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5)*Log[1 + 5

*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5])/(x + 5*x^2 + 6*x^3 - 9*x^4 + 3*x^5 + x^6),x]

[Out]

Log[x]*Log[1 + 5*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5] - Log[x]*Log[1 - x/Root[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4

+ #1^5 & , 1, 0]] - Log[x]*Log[1 - x/Root[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , 2, 0]] - Log[x]*Log[1

- x/Root[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , 3, 0]] - Log[x]*Log[1 - x/Root[1 + 5*#1 + 6*#1^2 - 9*

#1^3 + 3*#1^4 + #1^5 & , 4, 0]] - Log[x]*Log[1 - x/Root[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , 5, 0]]

- PolyLog[2, x/Root[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , 1, 0]] - PolyLog[2, x/Root[1 + 5*#1 + 6*#1^

2 - 9*#1^3 + 3*#1^4 + #1^5 & , 2, 0]] - PolyLog[2, x/Root[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , 3, 0]

] - PolyLog[2, x/Root[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , 4, 0]] - PolyLog[2, x/Root[1 + 5*#1 + 6*#

1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , 5, 0]] + ((5 + 12*x - 27*x^2 + 12*x^3 + 5*x^4)*RootSum[1 + 5*#1 + 6*#1^2 - 9*

#1^3 + 3*#1^4 + #1^5 & , (-Log[x]^2 + 2*Log[x]*Log[1 - x/#1] + 2*PolyLog[2, x/#1])/(5 + 12*#1 - 27*#1^2 + 12*#

1^3 + 5*#1^4) & ])/2 - (6 - 27*x + 18*x^2 + 10*x^3)*RootSum[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , (-(

x*Log[x]^2) + 2*x*Log[x]*Log[1 - x/#1] + 2*x*PolyLog[2, x/#1] - 2*Log[1 - x/#1]*#1 + 2*Log[x - #1]*#1 - 2*Log[

x]*Log[x - #1]*#1 + 2*Log[x - #1]*Log[x/#1]*#1 + 2*PolyLog[2, 1 - x/#1]*#1)/(5 + 12*#1 - 27*#1^2 + 12*#1^3 + 5

*#1^4) & ] - (27*RootSum[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , (-2*x^2*Log[x]^2 + 4*x^2*Log[x]*Log[1

- x/#1] + 4*x^2*PolyLog[2, x/#1] - 4*x*#1 + 4*x*Log[x]*#1 - 8*x*Log[1 - x/#1]*#1 + 8*x*Log[x - #1]*#1 - 8*x*Lo

g[x]*Log[x - #1]*#1 + 8*x*Log[x - #1]*Log[x/#1]*#1 + 8*x*PolyLog[2, 1 - x/#1]*#1 + 3*#1^2 + 6*Log[1 - x/#1]*#1

^2 - 8*Log[x]*Log[1 - x/#1]*#1^2 - 6*Log[x - #1]*#1^2 + 12*Log[x]*Log[x - #1]*#1^2 - 12*Log[x - #1]*Log[x/#1]*

#1^2 - 12*PolyLog[2, 1 - x/#1]*#1^2 - 8*PolyLog[2, x/#1]*#1^2)/(5 + 12*#1 - 27*#1^2 + 12*#1^3 + 5*#1^4) & ])/4

+ 9*x*RootSum[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , (-2*x^2*Log[x]^2 + 4*x^2*Log[x]*Log[1 - x/#1] +

4*x^2*PolyLog[2, x/#1] - 4*x*#1 + 4*x*Log[x]*#1 - 8*x*Log[1 - x/#1]*#1 + 8*x*Log[x - #1]*#1 - 8*x*Log[x]*Log[x

- #1]*#1 + 8*x*Log[x - #1]*Log[x/#1]*#1 + 8*x*PolyLog[2, 1 - x/#1]*#1 + 3*#1^2 + 6*Log[1 - x/#1]*#1^2 - 8*Log

[x]*Log[1 - x/#1]*#1^2 - 6*Log[x - #1]*#1^2 + 12*Log[x]*Log[x - #1]*#1^2 - 12*Log[x - #1]*Log[x/#1]*#1^2 - 12*

PolyLog[2, 1 - x/#1]*#1^2 - 8*PolyLog[2, x/#1]*#1^2)/(5 + 12*#1 - 27*#1^2 + 12*#1^3 + 5*#1^4) & ] + (15*x^2*Ro

otSum[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , (-2*x^2*Log[x]^2 + 4*x^2*Log[x]*Log[1 - x/#1] + 4*x^2*Pol

yLog[2, x/#1] - 4*x*#1 + 4*x*Log[x]*#1 - 8*x*Log[1 - x/#1]*#1 + 8*x*Log[x - #1]*#1 - 8*x*Log[x]*Log[x - #1]*#1

+ 8*x*Log[x - #1]*Log[x/#1]*#1 + 8*x*PolyLog[2, 1 - x/#1]*#1 + 3*#1^2 + 6*Log[1 - x/#1]*#1^2 - 8*Log[x]*Log[1

- x/#1]*#1^2 - 6*Log[x - #1]*#1^2 + 12*Log[x]*Log[x - #1]*#1^2 - 12*Log[x - #1]*Log[x/#1]*#1^2 - 12*PolyLog[2

, 1 - x/#1]*#1^2 - 8*PolyLog[2, x/#1]*#1^2)/(5 + 12*#1 - 27*#1^2 + 12*#1^3 + 5*#1^4) & ])/2 - 3*RootSum[1 + 5*

#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5 & , (-2*x^3*Log[x]^2 + 4*x^3*Log[x]*Log[1 - x/#1] + 4*x^3*PolyLog[2, x/#1

] - 11*x^2*#1 + 10*x^2*Log[x]*#1 - 12*x^2*Log[1 - x/#1]*#1 + 12*x^2*Log[x - #1]*#1 - 12*x^2*Log[x]*Log[x - #1]

*#1 + 12*x^2*Log[x - #1]*Log[x/#1]*#1 + 12*x^2*PolyLog[2, 1 - x/#1]*#1 + 13*x*#1^2 - 4*x*Log[x]*#1^2 + 18*x*Lo

g[1 - x/#1]*#1^2 - 24*x*Log[x]*Log[1 - x/#1]*#1^2 - 18*x*Log[x - #1]*#1^2 + 36*x*Log[x]*Log[x - #1]*#1^2 - 36*

x*Log[x - #1]*Log[x/#1]*#1^2 - 36*x*PolyLog[2, 1 - x/#1]*#1^2 - 24*x*PolyLog[2, x/#1]*#1^2 - 27*#1^3 + 18*Log[

1 - x/#1]*#1^3 + 24*Log[x]*Log[1 - x/#1]*#1^3 - 24*Log[x - #1]*#1^3 - 28*Log[x]*Log[x - #1]*#1^3 + 6*Log[2*(x

- #1)]*#1^3 + 12*Log[1 - x/#1]*Log[x/#1]*#1^3 + 28*Log[x - #1]*Log[x/#1]*#1^3 + 40*PolyLog[2, 1 - x/#1]*#1^3 +

36*PolyLog[2, x/#1]*#1^3)/(5 + 12*#1 - 27*#1^2 + 12*#1^3 + 5*#1^4) & ] - 5*x*RootSum[1 + 5*#1 + 6*#1^2 - 9*#1

^3 + 3*#1^4 + #1^5 & , (-2*x^3*Log[x]^2 + 4*x^3*Log[x]*Log[1 - x/#1] + 4*x^3*PolyLog[2, x/#1] - 11*x^2*#1 + 10

*x^2*Log[x]*#1 - 12*x^2*Log[1 - x/#1]*#1 + 12*x^2*Log[x - #1]*#1 - 12*x^2*Log[x]*Log[x - #1]*#1 + 12*x^2*Log[x

- #1]*Log[x/#1]*#1 + 12*x^2*PolyLog[2, 1 - x/#1]*#1 + 13*x*#1^2 - 4*x*Log[x]*#1^2 + 18*x*Log[1 - x/#1]*#1^2 -

24*x*Log[x]*Log[1 - x/#1]*#1^2 - 18*x*Log[x - #1]*#1^2 + 36*x*Log[x]*Log[x - #1]*#1^2 - 36*x*Log[x - #1]*Log[

x/#1]*#1^2 - 36*x*PolyLog[2, 1 - x/#1]*#1^2 - 24*x*PolyLog[2, x/#1]*#1^2 - 27*#1^3 + 18*Log[1 - x/#1]*#1^3 + 2

4*Log[x]*Log[1 - x/#1]*#1^3 - 24*Log[x - #1]*#1^3 - 28*Log[x]*Log[x - #1]*#1^3 + 6*Log[2*(x - #1)]*#1^3 + 12*L

og[1 - x/#1]*Log[x/#1]*#1^3 + 28*Log[x - #1]*Log[x/#1]*#1^3 + 40*PolyLog[2, 1 - x/#1]*#1^3 + 36*PolyLog[2, x/#

1]*#1^3)/(5 + 12*#1 - 27*#1^2 + 12*#1^3 + 5*#1^4) & ] + (5*RootSum[1 + 5*#1 + 6*#1^2 - 9*#1^3 + 3*#1^4 + #1^5

& , (-18*x^4*Log[x]^2 + 36*x^4*Log[x]*Log[1 - x/#1] + 36*x^4*PolyLog[2, x/#1] - 184*x^3*#1 + 156*x^3*Log[x]*#1

- 144*x^3*Log[1 - x/#1]*#1 + 144*x^3*Log[x - #1]*#1 - 144*x^3*Log[x]*Log[x - #1]*#1 + 144*x^3*Log[x - #1]*Log

[x/#1]*#1 + 144*x^3*PolyLog[2, 1 - x/#1]*#1 + 297*x^2*#1^2 - 126*x^2*Log[x]*#1^2 + 324*x^2*Log[1 - x/#1]*#1^2

- 432*x^2*Log[x]*Log[1 - x/#1]*#1^2 - 324*x^2*Log[x - #1]*#1^2 + 648*x^2*Log[x]*Log[x - #1]*#1^2 - 648*x^2*Log

[x - #1]*Log[x/#1]*#1^2 - 648*x^2*PolyLog[2, 1 - x/#1]*#1^2 - 432*x^2*PolyLog[2, x/#1]*#1^2 - 1008*x*#1^3 + 36

*x*Log[x]*#1^3 + 648*x*Log[1 - x/#1]*#1^3 + 864*x*Log[x]*Log[1 - x/#1]*#1^3 - 864*x*Log[x - #1]*#1^3 - 1008*x*

Log[x]*Log[x - #1]*#1^3 + 216*x*Log[2*(x - #1)]*#1^3 + 432*x*Log[1 - x/#1]*Log[x/#1]*#1^3 + 1008*x*Log[x - #1]

*Log[x/#1]*#1^3 + 1440*x*PolyLog[2, 1 - x/#1]*#1^3 + 1296*x*PolyLog[2, x/#1]*#1^3 + 2162*#1^4 - 2052*Log[1 - x

/#1]*#1^4 - 504*Log[x]*Log[1 - x/#1]*#1^4 + 432*Log[-1 + x/#1]*#1^4 + 2112*Log[x - #1]*#1^4 + 540*Log[x]*Log[x

- #1]*#1^4 - 540*Log[2*(x - #1)]*#1^4 + 48*Log[3*(x - #1)]*#1^4 - 936*Log[1 - x/#1]*Log[x/#1]*#1^4 - 540*Log[

x - #1]*Log[x/#1]*#1^4 - 1476*PolyLog[2, 1 - x/#1]*#1^4 - 1440*PolyLog[2, x/#1]*#1^4)/(5 + 12*#1 - 27*#1^2 + 1

2*#1^3 + 5*#1^4) & ])/36

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fricas [A] time = 0.71, size = 27, normalized size = 1.29 \begin {gather*} \log \left (x^{5} + 3 \, x^{4} - 9 \, x^{3} + 6 \, x^{2} + 5 \, x + 1\right ) \log \relax (x) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+3*x^4-9*x^3+6*x^2+5*x+1)*log(x^5+3*x^4-9*x^3+6*x^2+5*x+1)+(5*x^5+12*x^4-27*x^3+12*x^2+5*x)*log

(x))/(x^6+3*x^5-9*x^4+6*x^3+5*x^2+x),x, algorithm="fricas")

[Out]

log(x^5 + 3*x^4 - 9*x^3 + 6*x^2 + 5*x + 1)*log(x)

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giac [A] time = 0.25, size = 27, normalized size = 1.29 \begin {gather*} \log \left (x^{5} + 3 \, x^{4} - 9 \, x^{3} + 6 \, x^{2} + 5 \, x + 1\right ) \log \relax (x) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+3*x^4-9*x^3+6*x^2+5*x+1)*log(x^5+3*x^4-9*x^3+6*x^2+5*x+1)+(5*x^5+12*x^4-27*x^3+12*x^2+5*x)*log

(x))/(x^6+3*x^5-9*x^4+6*x^3+5*x^2+x),x, algorithm="giac")

[Out]

log(x^5 + 3*x^4 - 9*x^3 + 6*x^2 + 5*x + 1)*log(x)

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maple [A] time = 0.04, size = 28, normalized size = 1.33


method	result	size

default	\(\ln \relax (x ) \ln \left (x^{5}+3 x^{4}-9 x^{3}+6 x^{2}+5 x +1\right )\)	\(28\)
risch	\(\ln \relax (x ) \ln \left (x^{5}+3 x^{4}-9 x^{3}+6 x^{2}+5 x +1\right )\)	\(28\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5+3*x^4-9*x^3+6*x^2+5*x+1)*ln(x^5+3*x^4-9*x^3+6*x^2+5*x+1)+(5*x^5+12*x^4-27*x^3+12*x^2+5*x)*ln(x))/(x^

6+3*x^5-9*x^4+6*x^3+5*x^2+x),x,method=_RETURNVERBOSE)

[Out]

ln(x)*ln(x^5+3*x^4-9*x^3+6*x^2+5*x+1)

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maxima [A] time = 0.41, size = 27, normalized size = 1.29 \begin {gather*} \log \left (x^{5} + 3 \, x^{4} - 9 \, x^{3} + 6 \, x^{2} + 5 \, x + 1\right ) \log \relax (x) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+3*x^4-9*x^3+6*x^2+5*x+1)*log(x^5+3*x^4-9*x^3+6*x^2+5*x+1)+(5*x^5+12*x^4-27*x^3+12*x^2+5*x)*log

(x))/(x^6+3*x^5-9*x^4+6*x^3+5*x^2+x),x, algorithm="maxima")

[Out]

log(x^5 + 3*x^4 - 9*x^3 + 6*x^2 + 5*x + 1)*log(x)

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mupad [B] time = 1.80, size = 27, normalized size = 1.29 \begin {gather*} \ln \left (x^5+3\,x^4-9\,x^3+6\,x^2+5\,x+1\right )\,\ln \relax (x) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((log(5*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5 + 1)*(5*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5 + 1) + log(x)*(5*x + 12*x^2

- 27*x^3 + 12*x^4 + 5*x^5))/(x + 5*x^2 + 6*x^3 - 9*x^4 + 3*x^5 + x^6),x)

[Out]

log(5*x + 6*x^2 - 9*x^3 + 3*x^4 + x^5 + 1)*log(x)

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sympy [A] time = 0.38, size = 27, normalized size = 1.29 \begin {gather*} \log {\relax (x )} \log {\left (x^{5} + 3 x^{4} - 9 x^{3} + 6 x^{2} + 5 x + 1 \right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5+3*x**4-9*x**3+6*x**2+5*x+1)*ln(x**5+3*x**4-9*x**3+6*x**2+5*x+1)+(5*x**5+12*x**4-27*x**3+12*x*

*2+5*x)*ln(x))/(x**6+3*x**5-9*x**4+6*x**3+5*x**2+x),x)

[Out]

log(x)*log(x**5 + 3*x**4 - 9*x**3 + 6*x**2 + 5*x + 1)

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