3.1.58 \(\int x^9 \log ^{11}(x) \, dx\) [58]

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

[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

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2342, 2341} \begin {gather*} -\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} \end {gather*}

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 2341

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 2342

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]

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 = 0.00, size = 127, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 104, normalized size = 0.82

method	result	size
default	\(-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \ln \left (x \right )}{15625000}-\frac {6237 x^{10} \ln \left (x \right )^{2}}{3125000}+\frac {2079 x^{10} \ln \left (x \right )^{3}}{312500}-\frac {2079 x^{10} \ln \left (x \right )^{4}}{125000}+\frac {2079 x^{10} \ln \left (x \right )^{5}}{62500}-\frac {693 x^{10} \ln \left (x \right )^{6}}{12500}+\frac {99 x^{10} \ln \left (x \right )^{7}}{1250}-\frac {99 x^{10} \ln \left (x \right )^{8}}{1000}+\frac {11 x^{10} \ln \left (x \right )^{9}}{100}-\frac {11 x^{10} \ln \left (x \right )^{10}}{100}+\frac {x^{10} \ln \left (x \right )^{11}}{10}\)	\(104\)
risch	\(-\frac {6237 x^{10}}{156250000}+\frac {6237 x^{10} \ln \left (x \right )}{15625000}-\frac {6237 x^{10} \ln \left (x \right )^{2}}{3125000}+\frac {2079 x^{10} \ln \left (x \right )^{3}}{312500}-\frac {2079 x^{10} \ln \left (x \right )^{4}}{125000}+\frac {2079 x^{10} \ln \left (x \right )^{5}}{62500}-\frac {693 x^{10} \ln \left (x \right )^{6}}{12500}+\frac {99 x^{10} \ln \left (x \right )^{7}}{1250}-\frac {99 x^{10} \ln \left (x \right )^{8}}{1000}+\frac {11 x^{10} \ln \left (x \right )^{9}}{100}-\frac {11 x^{10} \ln \left (x \right )^{10}}{100}+\frac {x^{10} \ln \left (x \right )^{11}}{10}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*ln(x)^11,x,method=_RETURNVERBOSE)

[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

________________________________________________________________________________________

Maxima [A]
time = 1.80, size = 71, normalized size = 0.56 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 1.08, size = 103, normalized size = 0.81 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 0.15, size = 133, normalized size = 1.05 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.62, size = 103, normalized size = 0.81 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.18, size = 71, normalized size = 0.56 \begin {gather*} \frac {6237\,x^{10}\,\left (\frac {15625000\,{\ln \left (x\right )}^{11}}{6237}-\frac {1562500\,{\ln \left (x\right )}^{10}}{567}+\frac {1562500\,{\ln \left (x\right )}^9}{567}-\frac {156250\,{\ln \left (x\right )}^8}{63}+\frac {125000\,{\ln \left (x\right )}^7}{63}-\frac {12500\,{\ln \left (x\right )}^6}{9}+\frac {2500\,{\ln \left (x\right )}^5}{3}-\frac {1250\,{\ln \left (x\right )}^4}{3}+\frac {500\,{\ln \left (x\right )}^3}{3}-50\,{\ln \left (x\right )}^2+10\,\ln \left (x\right )-1\right )}{156250000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6237*x^10*(10*log(x) - 50*log(x)^2 + (500*log(x)^3)/3 - (1250*log(x)^4)/3 + (2500*log(x)^5)/3 - (12500*log(x)
^6)/9 + (125000*log(x)^7)/63 - (156250*log(x)^8)/63 + (1562500*log(x)^9)/567 - (1562500*log(x)^10)/567 + (1562
5000*log(x)^11)/6237 - 1))/156250000

________________________________________________________________________________________