3.92.83 \(\int \frac {-80-800 x-3320 x^2-7400 x^3-9605 x^4-7360 x^5-3320 x^6-800 x^7-80 x^8+(400+3200 x+9960 x^2+14800 x^3+9605 x^4-3320 x^6-1600 x^7-240 x^8) \log (x)}{x^6} \, dx\)

Optimal. Leaf size=24 \[ 5 \left (9-\frac {x+\left (5+2 \left (\frac {1}{x}+x\right )\right )^4}{x}\right ) \log (x) \]

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Rubi [B]  time = 0.24, antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps used = 14, number of rules used = 4, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {14, 2357, 2295, 2304} \begin {gather*} -\frac {80 \log (x)}{x^5}-\frac {800 \log (x)}{x^4}-80 x^3 \log (x)-\frac {3320 \log (x)}{x^3}-800 x^2 \log (x)-\frac {7400 \log (x)}{x^2}-3320 x \log (x)-7360 \log (x)-\frac {9605 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-80 - 800*x - 3320*x^2 - 7400*x^3 - 9605*x^4 - 7360*x^5 - 3320*x^6 - 800*x^7 - 80*x^8 + (400 + 3200*x + 9
960*x^2 + 14800*x^3 + 9605*x^4 - 3320*x^6 - 1600*x^7 - 240*x^8)*Log[x])/x^6,x]

[Out]

-7360*Log[x] - (80*Log[x])/x^5 - (800*Log[x])/x^4 - (3320*Log[x])/x^3 - (7400*Log[x])/x^2 - (9605*Log[x])/x -
3320*x*Log[x] - 800*x^2*Log[x] - 80*x^3*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {5 \left (16+160 x+664 x^2+1480 x^3+1921 x^4+1472 x^5+664 x^6+160 x^7+16 x^8\right )}{x^6}-\frac {5 (2+x)^3 (1+2 x)^3 \left (-10-5 x+6 x^2\right ) \log (x)}{x^6}\right ) \, dx\\ &=-\left (5 \int \frac {16+160 x+664 x^2+1480 x^3+1921 x^4+1472 x^5+664 x^6+160 x^7+16 x^8}{x^6} \, dx\right )-5 \int \frac {(2+x)^3 (1+2 x)^3 \left (-10-5 x+6 x^2\right ) \log (x)}{x^6} \, dx\\ &=-\left (5 \int \left (664+\frac {16}{x^6}+\frac {160}{x^5}+\frac {664}{x^4}+\frac {1480}{x^3}+\frac {1921}{x^2}+\frac {1472}{x}+160 x+16 x^2\right ) \, dx\right )-5 \int \left (664 \log (x)-\frac {80 \log (x)}{x^6}-\frac {640 \log (x)}{x^5}-\frac {1992 \log (x)}{x^4}-\frac {2960 \log (x)}{x^3}-\frac {1921 \log (x)}{x^2}+320 x \log (x)+48 x^2 \log (x)\right ) \, dx\\ &=\frac {16}{x^5}+\frac {200}{x^4}+\frac {3320}{3 x^3}+\frac {3700}{x^2}+\frac {9605}{x}-3320 x-400 x^2-\frac {80 x^3}{3}-7360 \log (x)-240 \int x^2 \log (x) \, dx+400 \int \frac {\log (x)}{x^6} \, dx-1600 \int x \log (x) \, dx+3200 \int \frac {\log (x)}{x^5} \, dx-3320 \int \log (x) \, dx+9605 \int \frac {\log (x)}{x^2} \, dx+9960 \int \frac {\log (x)}{x^4} \, dx+14800 \int \frac {\log (x)}{x^3} \, dx\\ &=-7360 \log (x)-\frac {80 \log (x)}{x^5}-\frac {800 \log (x)}{x^4}-\frac {3320 \log (x)}{x^3}-\frac {7400 \log (x)}{x^2}-\frac {9605 \log (x)}{x}-3320 x \log (x)-800 x^2 \log (x)-80 x^3 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 59, normalized size = 2.46 \begin {gather*} -7360 \log (x)-\frac {80 \log (x)}{x^5}-\frac {800 \log (x)}{x^4}-\frac {3320 \log (x)}{x^3}-\frac {7400 \log (x)}{x^2}-\frac {9605 \log (x)}{x}-3320 x \log (x)-800 x^2 \log (x)-80 x^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-80 - 800*x - 3320*x^2 - 7400*x^3 - 9605*x^4 - 7360*x^5 - 3320*x^6 - 800*x^7 - 80*x^8 + (400 + 3200
*x + 9960*x^2 + 14800*x^3 + 9605*x^4 - 3320*x^6 - 1600*x^7 - 240*x^8)*Log[x])/x^6,x]

[Out]

-7360*Log[x] - (80*Log[x])/x^5 - (800*Log[x])/x^4 - (3320*Log[x])/x^3 - (7400*Log[x])/x^2 - (9605*Log[x])/x -
3320*x*Log[x] - 800*x^2*Log[x] - 80*x^3*Log[x]

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fricas [A]  time = 0.90, size = 47, normalized size = 1.96 \begin {gather*} -\frac {5 \, {\left (16 \, x^{8} + 160 \, x^{7} + 664 \, x^{6} + 1472 \, x^{5} + 1921 \, x^{4} + 1480 \, x^{3} + 664 \, x^{2} + 160 \, x + 16\right )} \log \relax (x)}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+400)*log(x)-80*x^8-800*x^7-3320*x^6-
7360*x^5-9605*x^4-7400*x^3-3320*x^2-800*x-80)/x^6,x, algorithm="fricas")

[Out]

-5*(16*x^8 + 160*x^7 + 664*x^6 + 1472*x^5 + 1921*x^4 + 1480*x^3 + 664*x^2 + 160*x + 16)*log(x)/x^5

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giac [A]  time = 0.15, size = 47, normalized size = 1.96 \begin {gather*} -5 \, {\left (16 \, x^{3} + 160 \, x^{2} + 664 \, x + \frac {1921 \, x^{4} + 1480 \, x^{3} + 664 \, x^{2} + 160 \, x + 16}{x^{5}}\right )} \log \relax (x) - 7360 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+400)*log(x)-80*x^8-800*x^7-3320*x^6-
7360*x^5-9605*x^4-7400*x^3-3320*x^2-800*x-80)/x^6,x, algorithm="giac")

[Out]

-5*(16*x^3 + 160*x^2 + 664*x + (1921*x^4 + 1480*x^3 + 664*x^2 + 160*x + 16)/x^5)*log(x) - 7360*log(x)

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maple [A]  time = 0.04, size = 48, normalized size = 2.00

method	result	size
risch	\(-\frac {5 \left (16 x^{8}+160 x^{7}+664 x^{6}+1921 x^{4}+1480 x^{3}+664 x^{2}+160 x +16\right ) \ln \relax (x )}{x^{5}}-7360 \ln \relax (x )\)	\(48\)
default	\(-800 x^{2} \ln \relax (x )-\frac {9605 \ln \relax (x )}{x}-7360 \ln \relax (x )-80 x^{3} \ln \relax (x )-\frac {7400 \ln \relax (x )}{x^{2}}-3320 x \ln \relax (x )-\frac {3320 \ln \relax (x )}{x^{3}}-\frac {800 \ln \relax (x )}{x^{4}}-\frac {80 \ln \relax (x )}{x^{5}}\)	\(60\)
norman	\(\frac {-7360 x^{5} \ln \relax (x )-800 x \ln \relax (x )-3320 x^{2} \ln \relax (x )-7400 x^{3} \ln \relax (x )-9605 x^{4} \ln \relax (x )-3320 x^{6} \ln \relax (x )-800 x^{7} \ln \relax (x )-80 x^{8} \ln \relax (x )-80 \ln \relax (x )}{x^{5}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+400)*ln(x)-80*x^8-800*x^7-3320*x^6-7360*x^
5-9605*x^4-7400*x^3-3320*x^2-800*x-80)/x^6,x,method=_RETURNVERBOSE)

[Out]

-5*(16*x^8+160*x^7+664*x^6+1921*x^4+1480*x^3+664*x^2+160*x+16)/x^5*ln(x)-7360*ln(x)

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maxima [B]  time = 0.35, size = 59, normalized size = 2.46 \begin {gather*} -80 \, x^{3} \log \relax (x) - 800 \, x^{2} \log \relax (x) - 3320 \, x \log \relax (x) - \frac {9605 \, \log \relax (x)}{x} - \frac {7400 \, \log \relax (x)}{x^{2}} - \frac {3320 \, \log \relax (x)}{x^{3}} - \frac {800 \, \log \relax (x)}{x^{4}} - \frac {80 \, \log \relax (x)}{x^{5}} - 7360 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-240*x^8-1600*x^7-3320*x^6+9605*x^4+14800*x^3+9960*x^2+3200*x+400)*log(x)-80*x^8-800*x^7-3320*x^6-
7360*x^5-9605*x^4-7400*x^3-3320*x^2-800*x-80)/x^6,x, algorithm="maxima")

[Out]

-80*x^3*log(x) - 800*x^2*log(x) - 3320*x*log(x) - 9605*log(x)/x - 7400*log(x)/x^2 - 3320*log(x)/x^3 - 800*log(
x)/x^4 - 80*log(x)/x^5 - 7360*log(x)

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mupad [B]  time = 7.35, size = 67, normalized size = 2.79 \begin {gather*} -\frac {800\,x^2\,\ln \relax (x)+3320\,x^3\,\ln \relax (x)+7400\,x^4\,\ln \relax (x)+9605\,x^5\,\ln \relax (x)+7360\,x^6\,\ln \relax (x)+3320\,x^7\,\ln \relax (x)+800\,x^8\,\ln \relax (x)+80\,x^9\,\ln \relax (x)+80\,x\,\ln \relax (x)}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(800*x - log(x)*(3200*x + 9960*x^2 + 14800*x^3 + 9605*x^4 - 3320*x^6 - 1600*x^7 - 240*x^8 + 400) + 3320*x
^2 + 7400*x^3 + 9605*x^4 + 7360*x^5 + 3320*x^6 + 800*x^7 + 80*x^8 + 80)/x^6,x)

[Out]

-(800*x^2*log(x) + 3320*x^3*log(x) + 7400*x^4*log(x) + 9605*x^5*log(x) + 7360*x^6*log(x) + 3320*x^7*log(x) + 8
00*x^8*log(x) + 80*x^9*log(x) + 80*x*log(x))/x^6

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sympy [B]  time = 0.20, size = 48, normalized size = 2.00 \begin {gather*} - 7360 \log {\relax (x )} + \frac {\left (- 80 x^{8} - 800 x^{7} - 3320 x^{6} - 9605 x^{4} - 7400 x^{3} - 3320 x^{2} - 800 x - 80\right ) \log {\relax (x )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-240*x**8-1600*x**7-3320*x**6+9605*x**4+14800*x**3+9960*x**2+3200*x+400)*ln(x)-80*x**8-800*x**7-33
20*x**6-7360*x**5-9605*x**4-7400*x**3-3320*x**2-800*x-80)/x**6,x)

[Out]

-7360*log(x) + (-80*x**8 - 800*x**7 - 3320*x**6 - 9605*x**4 - 7400*x**3 - 3320*x**2 - 800*x - 80)*log(x)/x**5

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