3.83.14 \(\int \frac {-6+22 x-24 x^2+36 x^3-18 x \log (x)}{4 x^3-24 x^4+36 x^5+(-12 x^2+36 x^3) \log (x)+9 x \log ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ -\frac {1}{x-\frac {\log (x)}{\frac {2}{3}-2 x}} \]

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Rubi [F]  time = 0.66, 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 {-6+22 x-24 x^2+36 x^3-18 x \log (x)}{4 x^3-24 x^4+36 x^5+\left (-12 x^2+36 x^3\right ) \log (x)+9 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 + 22*x - 24*x^2 + 36*x^3 - 18*x*Log[x])/(4*x^3 - 24*x^4 + 36*x^5 + (-12*x^2 + 36*x^3)*Log[x] + 9*x*Log
[x]^2),x]

[Out]

22*Defer[Int][(-2*x + 6*x^2 + 3*Log[x])^(-2), x] - 6*Defer[Int][1/(x*(-2*x + 6*x^2 + 3*Log[x])^2), x] - 36*Def
er[Int][x/(-2*x + 6*x^2 + 3*Log[x])^2, x] + 72*Defer[Int][x^2/(-2*x + 6*x^2 + 3*Log[x])^2, x] - 6*Defer[Int][(
-2*x + 6*x^2 + 3*Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-3+11 x-12 x^2+18 x^3-9 x \log (x)\right )}{x (2 x (-1+3 x)+3 \log (x))^2} \, dx\\ &=2 \int \frac {-3+11 x-12 x^2+18 x^3-9 x \log (x)}{x (2 x (-1+3 x)+3 \log (x))^2} \, dx\\ &=2 \int \left (\frac {-3+11 x-18 x^2+36 x^3}{x \left (-2 x+6 x^2+3 \log (x)\right )^2}-\frac {3}{-2 x+6 x^2+3 \log (x)}\right ) \, dx\\ &=2 \int \frac {-3+11 x-18 x^2+36 x^3}{x \left (-2 x+6 x^2+3 \log (x)\right )^2} \, dx-6 \int \frac {1}{-2 x+6 x^2+3 \log (x)} \, dx\\ &=2 \int \left (\frac {11}{\left (-2 x+6 x^2+3 \log (x)\right )^2}-\frac {3}{x \left (-2 x+6 x^2+3 \log (x)\right )^2}-\frac {18 x}{\left (-2 x+6 x^2+3 \log (x)\right )^2}+\frac {36 x^2}{\left (-2 x+6 x^2+3 \log (x)\right )^2}\right ) \, dx-6 \int \frac {1}{-2 x+6 x^2+3 \log (x)} \, dx\\ &=-\left (6 \int \frac {1}{x \left (-2 x+6 x^2+3 \log (x)\right )^2} \, dx\right )-6 \int \frac {1}{-2 x+6 x^2+3 \log (x)} \, dx+22 \int \frac {1}{\left (-2 x+6 x^2+3 \log (x)\right )^2} \, dx-36 \int \frac {x}{\left (-2 x+6 x^2+3 \log (x)\right )^2} \, dx+72 \int \frac {x^2}{\left (-2 x+6 x^2+3 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 22, normalized size = 1.16 \begin {gather*} \frac {2 (1-3 x)}{-2 x+6 x^2+3 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 22*x - 24*x^2 + 36*x^3 - 18*x*Log[x])/(4*x^3 - 24*x^4 + 36*x^5 + (-12*x^2 + 36*x^3)*Log[x] + 9
*x*Log[x]^2),x]

[Out]

(2*(1 - 3*x))/(-2*x + 6*x^2 + 3*Log[x])

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fricas [A]  time = 1.02, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, {\left (3 \, x - 1\right )}}{6 \, x^{2} - 2 \, x + 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x*log(x)+36*x^3-24*x^2+22*x-6)/(9*x*log(x)^2+(36*x^3-12*x^2)*log(x)+36*x^5-24*x^4+4*x^3),x, alg
orithm="fricas")

[Out]

-2*(3*x - 1)/(6*x^2 - 2*x + 3*log(x))

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giac [A]  time = 0.13, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, {\left (3 \, x - 1\right )}}{6 \, x^{2} - 2 \, x + 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x*log(x)+36*x^3-24*x^2+22*x-6)/(9*x*log(x)^2+(36*x^3-12*x^2)*log(x)+36*x^5-24*x^4+4*x^3),x, alg
orithm="giac")

[Out]

-2*(3*x - 1)/(6*x^2 - 2*x + 3*log(x))

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maple [A]  time = 0.04, size = 22, normalized size = 1.16

method	result	size
norman	\(\frac {-6 x +2}{6 x^{2}+3 \ln \relax (x )-2 x}\)	\(22\)
risch	\(-\frac {2 \left (3 x -1\right )}{6 x^{2}+3 \ln \relax (x )-2 x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*x*ln(x)+36*x^3-24*x^2+22*x-6)/(9*x*ln(x)^2+(36*x^3-12*x^2)*ln(x)+36*x^5-24*x^4+4*x^3),x,method=_RETUR
NVERBOSE)

[Out]

(-6*x+2)/(6*x^2+3*ln(x)-2*x)

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maxima [A]  time = 0.40, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, {\left (3 \, x - 1\right )}}{6 \, x^{2} - 2 \, x + 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x*log(x)+36*x^3-24*x^2+22*x-6)/(9*x*log(x)^2+(36*x^3-12*x^2)*log(x)+36*x^5-24*x^4+4*x^3),x, alg
orithm="maxima")

[Out]

-2*(3*x - 1)/(6*x^2 - 2*x + 3*log(x))

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mupad [B]  time = 5.45, size = 22, normalized size = 1.16 \begin {gather*} -\frac {6\,x-2}{3\,\ln \relax (x)-2\,x+6\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x*log(x) - 22*x + 24*x^2 - 36*x^3 + 6)/(9*x*log(x)^2 - log(x)*(12*x^2 - 36*x^3) + 4*x^3 - 24*x^4 + 36
*x^5),x)

[Out]

-(6*x - 2)/(3*log(x) - 2*x + 6*x^2)

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sympy [A]  time = 0.12, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 - 6 x}{6 x^{2} - 2 x + 3 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*x*ln(x)+36*x**3-24*x**2+22*x-6)/(9*x*ln(x)**2+(36*x**3-12*x**2)*ln(x)+36*x**5-24*x**4+4*x**3),x
)

[Out]

(2 - 6*x)/(6*x**2 - 2*x + 3*log(x))

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