3.32.63 \(\int \frac {-13 x^6+11 x^7+(7 x^6-6 x^7) \log (x)}{1-2 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac {4 x^7 (2-\log (x))}{-(2-x)^2+x^2} \]

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Rubi [B]  time = 0.21, antiderivative size = 89, normalized size of antiderivative = 3.42, number of steps used = 15, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {27, 6742, 77, 2357, 2295, 2314, 31, 2304} \begin {gather*} 2 x^6+x^6 (-\log (x))+2 x^5-x^5 \log (x)+2 x^4-x^4 \log (x)+2 x^3-x^3 \log (x)+2 x^2-x^2 \log (x)+2 x-\frac {2}{1-x}+\frac {x \log (x)}{1-x}-x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-13*x^6 + 11*x^7 + (7*x^6 - 6*x^7)*Log[x])/(1 - 2*x + x^2),x]

[Out]

-2/(1 - x) + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 - x*Log[x] + (x*Log[x])/(1 - x) - x^2*Log[x] - x^3*Lo
g[x] - x^4*Log[x] - x^5*Log[x] - x^6*Log[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 31

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 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 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 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 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 {-13 x^6+11 x^7+\left (7 x^6-6 x^7\right ) \log (x)}{(-1+x)^2} \, dx\\ &=\int \left (\frac {x^6 (-13+11 x)}{(-1+x)^2}-\frac {x^6 (-7+6 x) \log (x)}{(-1+x)^2}\right ) \, dx\\ &=\int \frac {x^6 (-13+11 x)}{(-1+x)^2} \, dx-\int \frac {x^6 (-7+6 x) \log (x)}{(-1+x)^2} \, dx\\ &=\int \left (1+\frac {1}{1-x}-\frac {2}{(-1+x)^2}+3 x+5 x^2+7 x^3+9 x^4+11 x^5\right ) \, dx-\int \left (\log (x)-\frac {\log (x)}{(-1+x)^2}+2 x \log (x)+3 x^2 \log (x)+4 x^3 \log (x)+5 x^4 \log (x)+6 x^5 \log (x)\right ) \, dx\\ &=-\frac {2}{1-x}+x+\frac {3 x^2}{2}+\frac {5 x^3}{3}+\frac {7 x^4}{4}+\frac {9 x^5}{5}+\frac {11 x^6}{6}-\log (1-x)-2 \int x \log (x) \, dx-3 \int x^2 \log (x) \, dx-4 \int x^3 \log (x) \, dx-5 \int x^4 \log (x) \, dx-6 \int x^5 \log (x) \, dx-\int \log (x) \, dx+\int \frac {\log (x)}{(-1+x)^2} \, dx\\ &=-\frac {2}{1-x}+2 x+2 x^2+2 x^3+2 x^4+2 x^5+2 x^6-\log (1-x)-x \log (x)+\frac {x \log (x)}{1-x}-x^2 \log (x)-x^3 \log (x)-x^4 \log (x)-x^5 \log (x)-x^6 \log (x)+\int \frac {1}{-1+x} \, dx\\ &=-\frac {2}{1-x}+2 x+2 x^2+2 x^3+2 x^4+2 x^5+2 x^6-x \log (x)+\frac {x \log (x)}{1-x}-x^2 \log (x)-x^3 \log (x)-x^4 \log (x)-x^5 \log (x)-x^6 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 23, normalized size = 0.88 \begin {gather*} \frac {2-2 x+2 x^7-x^7 \log (x)}{-1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-13*x^6 + 11*x^7 + (7*x^6 - 6*x^7)*Log[x])/(1 - 2*x + x^2),x]

[Out]

(2 - 2*x + 2*x^7 - x^7*Log[x])/(-1 + x)

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fricas [A]  time = 0.69, size = 23, normalized size = 0.88 \begin {gather*} -\frac {x^{7} \log \relax (x) - 2 \, x^{7} + 2 \, x - 2}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^7+7*x^6)*log(x)+11*x^7-13*x^6)/(x^2-2*x+1),x, algorithm="fricas")

[Out]

-(x^7*log(x) - 2*x^7 + 2*x - 2)/(x - 1)

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giac [B]  time = 0.22, size = 66, normalized size = 2.54 \begin {gather*} 2 \, x^{6} + 2 \, x^{5} + 2 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} - {\left (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + \frac {1}{x - 1}\right )} \log \relax (x) + 2 \, x + \frac {2}{x - 1} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^7+7*x^6)*log(x)+11*x^7-13*x^6)/(x^2-2*x+1),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.64, size = 20, normalized size = 0.77

method	result	size
norman	\(\frac {2 x^{7}-x^{7} \ln \relax (x )}{x -1}\)	\(20\)
risch	\(-\frac {\left (x^{7}-x +1\right ) \ln \relax (x )}{x -1}-\frac {-2 x^{7}+x \ln \relax (x )-\ln \relax (x )+2 x -2}{x -1}\)	\(44\)
default	\(2 x^{6}+2 x^{5}+2 x^{4}+2 x^{3}+2 x^{2}+2 x +\frac {2}{x -1}-x^{6} \ln \relax (x )-x^{5} \ln \relax (x )-x^{4} \ln \relax (x )-x^{3} \ln \relax (x )-x^{2} \ln \relax (x )-x \ln \relax (x )-\frac {\ln \relax (x ) x}{x -1}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^7+7*x^6)*ln(x)+11*x^7-13*x^6)/(x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

(2*x^7-x^7*ln(x))/(x-1)

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maxima [B]  time = 0.76, size = 91, normalized size = 3.50 \begin {gather*} \frac {11}{6} \, x^{6} + \frac {9}{5} \, x^{5} + \frac {7}{4} \, x^{4} + \frac {5}{3} \, x^{3} + \frac {3}{2} \, x^{2} + x + \frac {10 \, x^{7} + 2 \, x^{6} + 3 \, x^{5} + 5 \, x^{4} + 10 \, x^{3} + 30 \, x^{2} - 60 \, {\left (x^{7} - x + 1\right )} \log \relax (x) - 60 \, x}{60 \, {\left (x - 1\right )}} + \frac {2}{x - 1} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^7+7*x^6)*log(x)+11*x^7-13*x^6)/(x^2-2*x+1),x, algorithm="maxima")

[Out]

11/6*x^6 + 9/5*x^5 + 7/4*x^4 + 5/3*x^3 + 3/2*x^2 + x + 1/60*(10*x^7 + 2*x^6 + 3*x^5 + 5*x^4 + 10*x^3 + 30*x^2
- 60*(x^7 - x + 1)*log(x) - 60*x)/(x - 1) + 2/(x - 1) - log(x)

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mupad [B]  time = 1.98, size = 14, normalized size = 0.54 \begin {gather*} -\frac {x^7\,\left (\ln \relax (x)-2\right )}{x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(7*x^6 - 6*x^7) - 13*x^6 + 11*x^7)/(x^2 - 2*x + 1),x)

[Out]

-(x^7*(log(x) - 2))/(x - 1)

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sympy [B]  time = 0.16, size = 49, normalized size = 1.88 \begin {gather*} 2 x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x - \log {\relax (x )} + \frac {\left (- x^{7} + x - 1\right ) \log {\relax (x )}}{x - 1} + \frac {2}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**7+7*x**6)*ln(x)+11*x**7-13*x**6)/(x**2-2*x+1),x)

[Out]

2*x**6 + 2*x**5 + 2*x**4 + 2*x**3 + 2*x**2 + 2*x - log(x) + (-x**7 + x - 1)*log(x)/(x - 1) + 2/(x - 1)

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