3.27.50 \(\int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac {25}{x}-\frac {1}{3} \log \left (e^{-6 e^{-x+x \log (x)}+2 x} x\right ) \]

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Rubi [F]  time = 0.08, 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 {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-75 - x - 2*x^2 + 6*E^(-x + x*Log[x])*x^2*Log[x])/(3*x^2),x]

[Out]

25/x - (2*x)/3 - Log[x]/3 + 2*Log[x]*Defer[Int][x^x/E^x, x] - 2*Defer[Int][Defer[Int][x^x/E^x, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{x^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {-75-x-2 x^2}{x^2}+6 e^{-x} x^x \log (x)\right ) \, dx\\ &=\frac {1}{3} \int \frac {-75-x-2 x^2}{x^2} \, dx+2 \int e^{-x} x^x \log (x) \, dx\\ &=\frac {1}{3} \int \left (-2-\frac {75}{x^2}-\frac {1}{x}\right ) \, dx-2 \int \frac {\int e^{-x} x^x \, dx}{x} \, dx+(2 \log (x)) \int e^{-x} x^x \, dx\\ &=\frac {25}{x}-\frac {2 x}{3}-\frac {\log (x)}{3}-2 \int \frac {\int e^{-x} x^x \, dx}{x} \, dx+(2 \log (x)) \int e^{-x} x^x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{3} \left (\frac {75}{x}-2 x+6 e^{-x} x^x-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-75 - x - 2*x^2 + 6*E^(-x + x*Log[x])*x^2*Log[x])/(3*x^2),x]

[Out]

(75/x - 2*x + (6*x^x)/E^x - Log[x])/3

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fricas [A]  time = 0.63, size = 28, normalized size = 0.90 \begin {gather*} -\frac {2 \, x^{2} - 6 \, x e^{\left (x \log \relax (x) - x\right )} + x \log \relax (x) - 75}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x^2*log(x)*exp(x*log(x)-x)-2*x^2-x-75)/x^2,x, algorithm="fricas")

[Out]

-1/3*(2*x^2 - 6*x*e^(x*log(x) - x) + x*log(x) - 75)/x

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giac [A]  time = 0.24, size = 28, normalized size = 0.90 \begin {gather*} -\frac {2 \, x^{2} - 6 \, x e^{\left (x \log \relax (x) - x\right )} + x \log \relax (x) - 75}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x^2*log(x)*exp(x*log(x)-x)-2*x^2-x-75)/x^2,x, algorithm="giac")

[Out]

-1/3*(2*x^2 - 6*x*e^(x*log(x) - x) + x*log(x) - 75)/x

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maple [A]  time = 0.02, size = 23, normalized size = 0.74

method	result	size
risch	\(-\frac {2 x}{3}-\frac {\ln \relax (x )}{3}+\frac {25}{x}+2 x^{x} {\mathrm e}^{-x}\)	\(23\)
default	\(-\frac {2 x}{3}-\frac {\ln \relax (x )}{3}+\frac {25}{x}+2 \,{\mathrm e}^{x \ln \relax (x )-x}\)	\(25\)
norman	\(\frac {25-\frac {x \ln \relax (x )}{3}-\frac {2 x^{2}}{3}+2 x \,{\mathrm e}^{x \ln \relax (x )-x}}{x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(6*x^2*ln(x)*exp(x*ln(x)-x)-2*x^2-x-75)/x^2,x,method=_RETURNVERBOSE)

[Out]

-2/3*x-1/3*ln(x)+25/x+2*x^x*exp(-x)

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maxima [A]  time = 0.35, size = 24, normalized size = 0.77 \begin {gather*} -\frac {2}{3} \, x + \frac {25}{x} + 2 \, e^{\left (x \log \relax (x) - x\right )} - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x^2*log(x)*exp(x*log(x)-x)-2*x^2-x-75)/x^2,x, algorithm="maxima")

[Out]

-2/3*x + 25/x + 2*e^(x*log(x) - x) - 1/3*log(x)

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mupad [B]  time = 1.54, size = 22, normalized size = 0.71 \begin {gather*} \frac {25}{x}-\frac {\ln \relax (x)}{3}-\frac {2\,x}{3}+2\,x^x\,{\mathrm {e}}^{-x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x/3 + (2*x^2)/3 - 2*x^2*exp(x*log(x) - x)*log(x) + 25)/x^2,x)

[Out]

25/x - log(x)/3 - (2*x)/3 + 2*x^x*exp(-x)

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sympy [A]  time = 0.27, size = 22, normalized size = 0.71 \begin {gather*} - \frac {2 x}{3} + 2 e^{x \log {\relax (x )} - x} - \frac {\log {\relax (x )}}{3} + \frac {25}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(6*x**2*ln(x)*exp(x*ln(x)-x)-2*x**2-x-75)/x**2,x)

[Out]

-2*x/3 + 2*exp(x*log(x) - x) - log(x)/3 + 25/x

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