3.58 \(\int x^9 \log ^{11}(x) \, dx\)

Optimal. Leaf size=127 \[ -\frac{6237 x^{10}}{156250000}+\frac{1}{10} x^{10} \log ^{11}(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{11}{100} x^{10} \log ^9(x)-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{2079 x^{10} \log ^4(x)}{125000}+\frac{2079 x^{10} \log ^3(x)}{312500}-\frac{6237 x^{10} \log ^2(x)}{3125000}+\frac{6237 x^{10} \log (x)}{15625000} \]

[Out]

(-6237*x^10)/156250000 + (6237*x^10*Log[x])/15625000 - (6237*x^10*Log[x]^2)/3125000 + (2079*x^10*Log[x]^3)/312
500 - (2079*x^10*Log[x]^4)/125000 + (2079*x^10*Log[x]^5)/62500 - (693*x^10*Log[x]^6)/12500 + (99*x^10*Log[x]^7
)/1250 - (99*x^10*Log[x]^8)/1000 + (11*x^10*Log[x]^9)/100 - (11*x^10*Log[x]^10)/100 + (x^10*Log[x]^11)/10

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Rubi [A]  time = 0.141166, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2305, 2304} \[ -\frac{6237 x^{10}}{156250000}+\frac{1}{10} x^{10} \log ^{11}(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{11}{100} x^{10} \log ^9(x)-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{2079 x^{10} \log ^4(x)}{125000}+\frac{2079 x^{10} \log ^3(x)}{312500}-\frac{6237 x^{10} \log ^2(x)}{3125000}+\frac{6237 x^{10} \log (x)}{15625000} \]

Antiderivative was successfully verified.

[In]

Int[x^9*Log[x]^11,x]

[Out]

(-6237*x^10)/156250000 + (6237*x^10*Log[x])/15625000 - (6237*x^10*Log[x]^2)/3125000 + (2079*x^10*Log[x]^3)/312
500 - (2079*x^10*Log[x]^4)/125000 + (2079*x^10*Log[x]^5)/62500 - (693*x^10*Log[x]^6)/12500 + (99*x^10*Log[x]^7
)/1250 - (99*x^10*Log[x]^8)/1000 + (11*x^10*Log[x]^9)/100 - (11*x^10*Log[x]^10)/100 + (x^10*Log[x]^11)/10

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 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
]

Rubi steps

\begin{align*} \int x^9 \log ^{11}(x) \, dx &=\frac{1}{10} x^{10} \log ^{11}(x)-\frac{11}{10} \int x^9 \log ^{10}(x) \, dx\\ &=-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)+\frac{11}{10} \int x^9 \log ^9(x) \, dx\\ &=\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)-\frac{99}{100} \int x^9 \log ^8(x) \, dx\\ &=-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)+\frac{99}{125} \int x^9 \log ^7(x) \, dx\\ &=\frac{99 x^{10} \log ^7(x)}{1250}-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)-\frac{693 \int x^9 \log ^6(x) \, dx}{1250}\\ &=-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)+\frac{2079 \int x^9 \log ^5(x) \, dx}{6250}\\ &=\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)-\frac{2079 \int x^9 \log ^4(x) \, dx}{12500}\\ &=-\frac{2079 x^{10} \log ^4(x)}{125000}+\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)+\frac{2079 \int x^9 \log ^3(x) \, dx}{31250}\\ &=\frac{2079 x^{10} \log ^3(x)}{312500}-\frac{2079 x^{10} \log ^4(x)}{125000}+\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)-\frac{6237 \int x^9 \log ^2(x) \, dx}{312500}\\ &=-\frac{6237 x^{10} \log ^2(x)}{3125000}+\frac{2079 x^{10} \log ^3(x)}{312500}-\frac{2079 x^{10} \log ^4(x)}{125000}+\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)+\frac{6237 \int x^9 \log (x) \, dx}{1562500}\\ &=-\frac{6237 x^{10}}{156250000}+\frac{6237 x^{10} \log (x)}{15625000}-\frac{6237 x^{10} \log ^2(x)}{3125000}+\frac{2079 x^{10} \log ^3(x)}{312500}-\frac{2079 x^{10} \log ^4(x)}{125000}+\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{11}{100} x^{10} \log ^9(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{1}{10} x^{10} \log ^{11}(x)\\ \end{align*}

Mathematica [A]  time = 0.0032175, size = 127, normalized size = 1. \[ -\frac{6237 x^{10}}{156250000}+\frac{1}{10} x^{10} \log ^{11}(x)-\frac{11}{100} x^{10} \log ^{10}(x)+\frac{11}{100} x^{10} \log ^9(x)-\frac{99 x^{10} \log ^8(x)}{1000}+\frac{99 x^{10} \log ^7(x)}{1250}-\frac{693 x^{10} \log ^6(x)}{12500}+\frac{2079 x^{10} \log ^5(x)}{62500}-\frac{2079 x^{10} \log ^4(x)}{125000}+\frac{2079 x^{10} \log ^3(x)}{312500}-\frac{6237 x^{10} \log ^2(x)}{3125000}+\frac{6237 x^{10} \log (x)}{15625000} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9*Log[x]^11,x]

[Out]

(-6237*x^10)/156250000 + (6237*x^10*Log[x])/15625000 - (6237*x^10*Log[x]^2)/3125000 + (2079*x^10*Log[x]^3)/312
500 - (2079*x^10*Log[x]^4)/125000 + (2079*x^10*Log[x]^5)/62500 - (693*x^10*Log[x]^6)/12500 + (99*x^10*Log[x]^7
)/1250 - (99*x^10*Log[x]^8)/1000 + (11*x^10*Log[x]^9)/100 - (11*x^10*Log[x]^10)/100 + (x^10*Log[x]^11)/10

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Maple [A]  time = 0.002, size = 104, normalized size = 0.8 \begin{align*} -{\frac{6237\,{x}^{10}}{156250000}}+{\frac{6237\,{x}^{10}\ln \left ( x \right ) }{15625000}}-{\frac{6237\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{2}}{3125000}}+{\frac{2079\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{3}}{312500}}-{\frac{2079\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{4}}{125000}}+{\frac{2079\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{5}}{62500}}-{\frac{693\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{6}}{12500}}+{\frac{99\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{7}}{1250}}-{\frac{99\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{8}}{1000}}+{\frac{11\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{9}}{100}}-{\frac{11\,{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{10}}{100}}+{\frac{{x}^{10} \left ( \ln \left ( x \right ) \right ) ^{11}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*ln(x)^11,x)

[Out]

-6237/156250000*x^10+6237/15625000*x^10*ln(x)-6237/3125000*x^10*ln(x)^2+2079/312500*x^10*ln(x)^3-2079/125000*x
^10*ln(x)^4+2079/62500*x^10*ln(x)^5-693/12500*x^10*ln(x)^6+99/1250*x^10*ln(x)^7-99/1000*x^10*ln(x)^8+11/100*x^
10*ln(x)^9-11/100*x^10*ln(x)^10+1/10*x^10*ln(x)^11

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Maxima [A]  time = 0.936561, size = 96, normalized size = 0.76 \begin{align*} \frac{1}{156250000} \,{\left (15625000 \, \log \left (x\right )^{11} - 17187500 \, \log \left (x\right )^{10} + 17187500 \, \log \left (x\right )^{9} - 15468750 \, \log \left (x\right )^{8} + 12375000 \, \log \left (x\right )^{7} - 8662500 \, \log \left (x\right )^{6} + 5197500 \, \log \left (x\right )^{5} - 2598750 \, \log \left (x\right )^{4} + 1039500 \, \log \left (x\right )^{3} - 311850 \, \log \left (x\right )^{2} + 62370 \, \log \left (x\right ) - 6237\right )} x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*log(x)^11,x, algorithm="maxima")

[Out]

1/156250000*(15625000*log(x)^11 - 17187500*log(x)^10 + 17187500*log(x)^9 - 15468750*log(x)^8 + 12375000*log(x)
^7 - 8662500*log(x)^6 + 5197500*log(x)^5 - 2598750*log(x)^4 + 1039500*log(x)^3 - 311850*log(x)^2 + 62370*log(x
) - 6237)*x^10

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Fricas [A]  time = 2.05302, size = 410, normalized size = 3.23 \begin{align*} \frac{1}{10} \, x^{10} \log \left (x\right )^{11} - \frac{11}{100} \, x^{10} \log \left (x\right )^{10} + \frac{11}{100} \, x^{10} \log \left (x\right )^{9} - \frac{99}{1000} \, x^{10} \log \left (x\right )^{8} + \frac{99}{1250} \, x^{10} \log \left (x\right )^{7} - \frac{693}{12500} \, x^{10} \log \left (x\right )^{6} + \frac{2079}{62500} \, x^{10} \log \left (x\right )^{5} - \frac{2079}{125000} \, x^{10} \log \left (x\right )^{4} + \frac{2079}{312500} \, x^{10} \log \left (x\right )^{3} - \frac{6237}{3125000} \, x^{10} \log \left (x\right )^{2} + \frac{6237}{15625000} \, x^{10} \log \left (x\right ) - \frac{6237}{156250000} \, x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*log(x)^11,x, algorithm="fricas")

[Out]

1/10*x^10*log(x)^11 - 11/100*x^10*log(x)^10 + 11/100*x^10*log(x)^9 - 99/1000*x^10*log(x)^8 + 99/1250*x^10*log(
x)^7 - 693/12500*x^10*log(x)^6 + 2079/62500*x^10*log(x)^5 - 2079/125000*x^10*log(x)^4 + 2079/312500*x^10*log(x
)^3 - 6237/3125000*x^10*log(x)^2 + 6237/15625000*x^10*log(x) - 6237/156250000*x^10

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Sympy [A]  time = 0.262542, size = 133, normalized size = 1.05 \begin{align*} \frac{x^{10} \log{\left (x \right )}^{11}}{10} - \frac{11 x^{10} \log{\left (x \right )}^{10}}{100} + \frac{11 x^{10} \log{\left (x \right )}^{9}}{100} - \frac{99 x^{10} \log{\left (x \right )}^{8}}{1000} + \frac{99 x^{10} \log{\left (x \right )}^{7}}{1250} - \frac{693 x^{10} \log{\left (x \right )}^{6}}{12500} + \frac{2079 x^{10} \log{\left (x \right )}^{5}}{62500} - \frac{2079 x^{10} \log{\left (x \right )}^{4}}{125000} + \frac{2079 x^{10} \log{\left (x \right )}^{3}}{312500} - \frac{6237 x^{10} \log{\left (x \right )}^{2}}{3125000} + \frac{6237 x^{10} \log{\left (x \right )}}{15625000} - \frac{6237 x^{10}}{156250000} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*ln(x)**11,x)

[Out]

x**10*log(x)**11/10 - 11*x**10*log(x)**10/100 + 11*x**10*log(x)**9/100 - 99*x**10*log(x)**8/1000 + 99*x**10*lo
g(x)**7/1250 - 693*x**10*log(x)**6/12500 + 2079*x**10*log(x)**5/62500 - 2079*x**10*log(x)**4/125000 + 2079*x**
10*log(x)**3/312500 - 6237*x**10*log(x)**2/3125000 + 6237*x**10*log(x)/15625000 - 6237*x**10/156250000

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Giac [A]  time = 1.07493, size = 139, normalized size = 1.09 \begin{align*} \frac{1}{10} \, x^{10} \log \left (x\right )^{11} - \frac{11}{100} \, x^{10} \log \left (x\right )^{10} + \frac{11}{100} \, x^{10} \log \left (x\right )^{9} - \frac{99}{1000} \, x^{10} \log \left (x\right )^{8} + \frac{99}{1250} \, x^{10} \log \left (x\right )^{7} - \frac{693}{12500} \, x^{10} \log \left (x\right )^{6} + \frac{2079}{62500} \, x^{10} \log \left (x\right )^{5} - \frac{2079}{125000} \, x^{10} \log \left (x\right )^{4} + \frac{2079}{312500} \, x^{10} \log \left (x\right )^{3} - \frac{6237}{3125000} \, x^{10} \log \left (x\right )^{2} + \frac{6237}{15625000} \, x^{10} \log \left (x\right ) - \frac{6237}{156250000} \, x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*log(x)^11,x, algorithm="giac")

[Out]

1/10*x^10*log(x)^11 - 11/100*x^10*log(x)^10 + 11/100*x^10*log(x)^9 - 99/1000*x^10*log(x)^8 + 99/1250*x^10*log(
x)^7 - 693/12500*x^10*log(x)^6 + 2079/62500*x^10*log(x)^5 - 2079/125000*x^10*log(x)^4 + 2079/312500*x^10*log(x
)^3 - 6237/3125000*x^10*log(x)^2 + 6237/15625000*x^10*log(x) - 6237/156250000*x^10