3.82.62 \(\int \frac {144 x+12 x^2+4 e^2 x^2+(-12 x^2-4 e^2 x^2) \log (x)}{324+(-108 x-36 e^2 x) \log (x)+(9 x^2+6 e^2 x^2+e^4 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {x^2}{\frac {9}{2}-\frac {1}{4} \left (3+e^2\right ) x \log (x)} \]

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Rubi [F]  time = 0.53, 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 {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(144*x + 12*x^2 + 4*E^2*x^2 + (-12*x^2 - 4*E^2*x^2)*Log[x])/(324 + (-108*x - 36*E^2*x)*Log[x] + (9*x^2 + 6
*E^2*x^2 + E^4*x^2)*Log[x]^2),x]

[Out]

72*Defer[Int][x/(18 - 3*(1 + E^2/3)*x*Log[x])^2, x] + 4*(3 + E^2)*Defer[Int][x^2/(18 - 3*(1 + E^2/3)*x*Log[x])
^2, x] + 4*Defer[Int][x/(18 - 3*(1 + E^2/3)*x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {144 x+\left (12+4 e^2\right ) x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {4 x \left (36+\left (3+e^2\right ) x-\left (3+e^2\right ) x \log (x)\right )}{\left (18-\left (3+e^2\right ) x \log (x)\right )^2} \, dx\\ &=4 \int \frac {x \left (36+\left (3+e^2\right ) x-\left (3+e^2\right ) x \log (x)\right )}{\left (18-\left (3+e^2\right ) x \log (x)\right )^2} \, dx\\ &=4 \int \left (\frac {x \left (18+\left (3+e^2\right ) x\right )}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2}+\frac {x}{18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)}\right ) \, dx\\ &=4 \int \frac {x \left (18+\left (3+e^2\right ) x\right )}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2} \, dx+4 \int \frac {x}{18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)} \, dx\\ &=4 \int \frac {x}{18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)} \, dx+4 \int \left (\frac {18 x}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2}+\frac {\left (3+e^2\right ) x^2}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2}\right ) \, dx\\ &=4 \int \frac {x}{18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)} \, dx+72 \int \frac {x}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2} \, dx+\left (4 \left (3+e^2\right )\right ) \int \frac {x^2}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 18, normalized size = 0.82 \begin {gather*} -\frac {4 x^2}{-18+\left (3+e^2\right ) x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(144*x + 12*x^2 + 4*E^2*x^2 + (-12*x^2 - 4*E^2*x^2)*Log[x])/(324 + (-108*x - 36*E^2*x)*Log[x] + (9*x
^2 + 6*E^2*x^2 + E^4*x^2)*Log[x]^2),x]

[Out]

(-4*x^2)/(-18 + (3 + E^2)*x*Log[x])

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fricas [A]  time = 0.92, size = 20, normalized size = 0.91 \begin {gather*} -\frac {4 \, x^{2}}{{\left (x e^{2} + 3 \, x\right )} \log \relax (x) - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2*exp(2)-12*x^2)*log(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2*exp(2)^2+6*x^2*exp(2)+9*x^2)*log(x)^
2+(-36*exp(2)*x-108*x)*log(x)+324),x, algorithm="fricas")

[Out]

-4*x^2/((x*e^2 + 3*x)*log(x) - 18)

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giac [A]  time = 0.25, size = 20, normalized size = 0.91 \begin {gather*} -\frac {4 \, x^{2}}{x e^{2} \log \relax (x) + 3 \, x \log \relax (x) - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2*exp(2)-12*x^2)*log(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2*exp(2)^2+6*x^2*exp(2)+9*x^2)*log(x)^
2+(-36*exp(2)*x-108*x)*log(x)+324),x, algorithm="giac")

[Out]

-4*x^2/(x*e^2*log(x) + 3*x*log(x) - 18)

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maple [A]  time = 0.09, size = 21, normalized size = 0.95

method	result	size
norman	\(-\frac {4 x^{2}}{x \,{\mathrm e}^{2} \ln \relax (x )+3 x \ln \relax (x )-18}\)	\(21\)
risch	\(-\frac {4 x^{2}}{x \,{\mathrm e}^{2} \ln \relax (x )+3 x \ln \relax (x )-18}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2*exp(2)-12*x^2)*ln(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2*exp(2)^2+6*x^2*exp(2)+9*x^2)*ln(x)^2+(-36*e
xp(2)*x-108*x)*ln(x)+324),x,method=_RETURNVERBOSE)

[Out]

-4*x^2/(x*exp(2)*ln(x)+3*x*ln(x)-18)

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maxima [A]  time = 0.40, size = 17, normalized size = 0.77 \begin {gather*} -\frac {4 \, x^{2}}{x {\left (e^{2} + 3\right )} \log \relax (x) - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2*exp(2)-12*x^2)*log(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2*exp(2)^2+6*x^2*exp(2)+9*x^2)*log(x)^
2+(-36*exp(2)*x-108*x)*log(x)+324),x, algorithm="maxima")

[Out]

-4*x^2/(x*(e^2 + 3)*log(x) - 18)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {144\,x-\ln \relax (x)\,\left (4\,x^2\,{\mathrm {e}}^2+12\,x^2\right )+4\,x^2\,{\mathrm {e}}^2+12\,x^2}{\left (6\,x^2\,{\mathrm {e}}^2+x^2\,{\mathrm {e}}^4+9\,x^2\right )\,{\ln \relax (x)}^2+\left (-108\,x-36\,x\,{\mathrm {e}}^2\right )\,\ln \relax (x)+324} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*x - log(x)*(4*x^2*exp(2) + 12*x^2) + 4*x^2*exp(2) + 12*x^2)/(log(x)^2*(6*x^2*exp(2) + x^2*exp(4) + 9*
x^2) - log(x)*(108*x + 36*x*exp(2)) + 324),x)

[Out]

int((144*x - log(x)*(4*x^2*exp(2) + 12*x^2) + 4*x^2*exp(2) + 12*x^2)/(log(x)^2*(6*x^2*exp(2) + x^2*exp(4) + 9*
x^2) - log(x)*(108*x + 36*x*exp(2)) + 324), x)

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sympy [A]  time = 0.17, size = 19, normalized size = 0.86 \begin {gather*} - \frac {4 x^{2}}{\left (3 x + x e^{2}\right ) \log {\relax (x )} - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2*exp(2)-12*x**2)*ln(x)+4*x**2*exp(2)+12*x**2+144*x)/((x**2*exp(2)**2+6*x**2*exp(2)+9*x**2)*
ln(x)**2+(-36*exp(2)*x-108*x)*ln(x)+324),x)

[Out]

-4*x**2/((3*x + x*exp(2))*log(x) - 18)

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