3.56.96 \(\int \frac {-729 x+162 x^3+729 x \log (x)}{729 x^3-972 x^4+432 x^5-64 x^6+(4374 x^2-3888 x^3+864 x^4) \log (x)+(8748 x-3888 x^2) \log ^2(x)+5832 \log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^2}{\left (-2 x-\frac {8 x^2}{9}+4 (x+\log (x))\right )^2} \]

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Rubi [F]  time = 0.55, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-729 x+162 x^3+729 x \log (x)}{729 x^3-972 x^4+432 x^5-64 x^6+\left (4374 x^2-3888 x^3+864 x^4\right ) \log (x)+\left (8748 x-3888 x^2\right ) \log ^2(x)+5832 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-729*x + 162*x^3 + 729*x*Log[x])/(729*x^3 - 972*x^4 + 432*x^5 - 64*x^6 + (4374*x^2 - 3888*x^3 + 864*x^4)*
Log[x] + (8748*x - 3888*x^2)*Log[x]^2 + 5832*Log[x]^3),x]

[Out]

729*Defer[Int][x/(-9*x + 4*x^2 - 18*Log[x])^3, x] + (729*Defer[Int][x^2/(-9*x + 4*x^2 - 18*Log[x])^3, x])/2 -
324*Defer[Int][x^3/(-9*x + 4*x^2 - 18*Log[x])^3, x] + (81*Defer[Int][x/(-9*x + 4*x^2 - 18*Log[x])^2, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {81 x \left (9-2 x^2-9 \log (x)\right )}{(x (-9+4 x)-18 \log (x))^3} \, dx\\ &=81 \int \frac {x \left (9-2 x^2-9 \log (x)\right )}{(x (-9+4 x)-18 \log (x))^3} \, dx\\ &=81 \int \left (-\frac {x \left (-18-9 x+8 x^2\right )}{2 \left (-9 x+4 x^2-18 \log (x)\right )^3}+\frac {x}{2 \left (-9 x+4 x^2-18 \log (x)\right )^2}\right ) \, dx\\ &=-\left (\frac {81}{2} \int \frac {x \left (-18-9 x+8 x^2\right )}{\left (-9 x+4 x^2-18 \log (x)\right )^3} \, dx\right )+\frac {81}{2} \int \frac {x}{\left (-9 x+4 x^2-18 \log (x)\right )^2} \, dx\\ &=-\left (\frac {81}{2} \int \left (-\frac {18 x}{\left (-9 x+4 x^2-18 \log (x)\right )^3}-\frac {9 x^2}{\left (-9 x+4 x^2-18 \log (x)\right )^3}+\frac {8 x^3}{\left (-9 x+4 x^2-18 \log (x)\right )^3}\right ) \, dx\right )+\frac {81}{2} \int \frac {x}{\left (-9 x+4 x^2-18 \log (x)\right )^2} \, dx\\ &=\frac {81}{2} \int \frac {x}{\left (-9 x+4 x^2-18 \log (x)\right )^2} \, dx-324 \int \frac {x^3}{\left (-9 x+4 x^2-18 \log (x)\right )^3} \, dx+\frac {729}{2} \int \frac {x^2}{\left (-9 x+4 x^2-18 \log (x)\right )^3} \, dx+729 \int \frac {x}{\left (-9 x+4 x^2-18 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 22, normalized size = 0.96 \begin {gather*} \frac {81 x^2}{4 \left (9 x-4 x^2+18 \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-729*x + 162*x^3 + 729*x*Log[x])/(729*x^3 - 972*x^4 + 432*x^5 - 64*x^6 + (4374*x^2 - 3888*x^3 + 864
*x^4)*Log[x] + (8748*x - 3888*x^2)*Log[x]^2 + 5832*Log[x]^3),x]

[Out]

(81*x^2)/(4*(9*x - 4*x^2 + 18*Log[x])^2)

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fricas [B]  time = 0.61, size = 42, normalized size = 1.83 \begin {gather*} \frac {81 \, x^{2}}{4 \, {\left (16 \, x^{4} - 72 \, x^{3} + 81 \, x^{2} - 36 \, {\left (4 \, x^{2} - 9 \, x\right )} \log \relax (x) + 324 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((729*x*log(x)+162*x^3-729*x)/(5832*log(x)^3+(-3888*x^2+8748*x)*log(x)^2+(864*x^4-3888*x^3+4374*x^2)*
log(x)-64*x^6+432*x^5-972*x^4+729*x^3),x, algorithm="fricas")

[Out]

81/4*x^2/(16*x^4 - 72*x^3 + 81*x^2 - 36*(4*x^2 - 9*x)*log(x) + 324*log(x)^2)

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giac [B]  time = 0.16, size = 94, normalized size = 4.09 \begin {gather*} \frac {81 \, {\left (8 \, x^{4} - 9 \, x^{3} - 18 \, x^{2}\right )}}{4 \, {\left (128 \, x^{6} - 720 \, x^{5} - 1152 \, x^{4} \log \relax (x) + 1008 \, x^{4} + 3888 \, x^{3} \log \relax (x) + 2592 \, x^{2} \log \relax (x)^{2} + 567 \, x^{3} - 324 \, x^{2} \log \relax (x) - 2916 \, x \log \relax (x)^{2} - 1458 \, x^{2} - 5832 \, x \log \relax (x) - 5832 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((729*x*log(x)+162*x^3-729*x)/(5832*log(x)^3+(-3888*x^2+8748*x)*log(x)^2+(864*x^4-3888*x^3+4374*x^2)*
log(x)-64*x^6+432*x^5-972*x^4+729*x^3),x, algorithm="giac")

[Out]

81/4*(8*x^4 - 9*x^3 - 18*x^2)/(128*x^6 - 720*x^5 - 1152*x^4*log(x) + 1008*x^4 + 3888*x^3*log(x) + 2592*x^2*log
(x)^2 + 567*x^3 - 324*x^2*log(x) - 2916*x*log(x)^2 - 1458*x^2 - 5832*x*log(x) - 5832*log(x)^2)

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maple [A]  time = 0.02, size = 21, normalized size = 0.91

method	result	size
risch	\(\frac {81 x^{2}}{4 \left (4 x^{2}-9 x -18 \ln \relax (x )\right )^{2}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((729*x*ln(x)+162*x^3-729*x)/(5832*ln(x)^3+(-3888*x^2+8748*x)*ln(x)^2+(864*x^4-3888*x^3+4374*x^2)*ln(x)-64*
x^6+432*x^5-972*x^4+729*x^3),x,method=_RETURNVERBOSE)

[Out]

81/4*x^2/(4*x^2-9*x-18*ln(x))^2

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maxima [B]  time = 0.43, size = 42, normalized size = 1.83 \begin {gather*} \frac {81 \, x^{2}}{4 \, {\left (16 \, x^{4} - 72 \, x^{3} + 81 \, x^{2} - 36 \, {\left (4 \, x^{2} - 9 \, x\right )} \log \relax (x) + 324 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((729*x*log(x)+162*x^3-729*x)/(5832*log(x)^3+(-3888*x^2+8748*x)*log(x)^2+(864*x^4-3888*x^3+4374*x^2)*
log(x)-64*x^6+432*x^5-972*x^4+729*x^3),x, algorithm="maxima")

[Out]

81/4*x^2/(16*x^4 - 72*x^3 + 81*x^2 - 36*(4*x^2 - 9*x)*log(x) + 324*log(x)^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {729\,x\,\ln \relax (x)-729\,x+162\,x^3}{{\ln \relax (x)}^2\,\left (8748\,x-3888\,x^2\right )+5832\,{\ln \relax (x)}^3+\ln \relax (x)\,\left (864\,x^4-3888\,x^3+4374\,x^2\right )+729\,x^3-972\,x^4+432\,x^5-64\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((729*x*log(x) - 729*x + 162*x^3)/(log(x)^2*(8748*x - 3888*x^2) + 5832*log(x)^3 + log(x)*(4374*x^2 - 3888*x
^3 + 864*x^4) + 729*x^3 - 972*x^4 + 432*x^5 - 64*x^6),x)

[Out]

int((729*x*log(x) - 729*x + 162*x^3)/(log(x)^2*(8748*x - 3888*x^2) + 5832*log(x)^3 + log(x)*(4374*x^2 - 3888*x
^3 + 864*x^4) + 729*x^3 - 972*x^4 + 432*x^5 - 64*x^6), x)

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sympy [A]  time = 0.16, size = 37, normalized size = 1.61 \begin {gather*} \frac {81 x^{2}}{64 x^{4} - 288 x^{3} + 324 x^{2} + \left (- 576 x^{2} + 1296 x\right ) \log {\relax (x )} + 1296 \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((729*x*ln(x)+162*x**3-729*x)/(5832*ln(x)**3+(-3888*x**2+8748*x)*ln(x)**2+(864*x**4-3888*x**3+4374*x*
*2)*ln(x)-64*x**6+432*x**5-972*x**4+729*x**3),x)

[Out]

81*x**2/(64*x**4 - 288*x**3 + 324*x**2 + (-576*x**2 + 1296*x)*log(x) + 1296*log(x)**2)

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