\(\int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx\) [2650]

Optimal result

Integrand size = 34, antiderivative size = 31 \[ \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx=\frac {25}{x}-\frac {1}{3} \log \left (e^{-6 e^{-x+x \log (x)}+2 x} x\right ) \]

[Out]

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

Rubi [F]

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

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx=\frac {1}{3} \left (\frac {75}{x}-2 x+6 e^{-x} x^x-\log (x)\right ) \]

[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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {2 x}{3}-\frac {\ln \left (x \right )}{3}+\frac {25}{x}+2 x^{x} {\mathrm e}^{-x}\)	\(23\)
default	\(-\frac {2 x}{3}-\frac {\ln \left (x \right )}{3}+\frac {25}{x}+2 \,{\mathrm e}^{x \ln \left (x \right )-x}\)	\(25\)
parts	\(-\frac {2 x}{3}-\frac {\ln \left (x \right )}{3}+\frac {25}{x}+2 \,{\mathrm e}^{x \ln \left (x \right )-x}\)	\(25\)
parallelrisch	\(-\frac {x \ln \left (x \right )+2 x^{2}-6 x \,{\mathrm e}^{\left (\ln \left (x \right )-1\right ) x}-75}{3 x}\)	\(27\)
norman	\(\frac {25-\frac {x \ln \left (x \right )}{3}-\frac {2 x^{2}}{3}+2 x \,{\mathrm e}^{x \ln \left (x \right )-x}}{x}\)	\(29\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx=-\frac {2 \, x^{2} - 6 \, x e^{\left (x \log \left (x\right ) - x\right )} + x \log \left (x\right ) - 75}{3 \, x} \]

[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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx=- \frac {2 x}{3} + 2 e^{x \log {\left (x \right )} - x} - \frac {\log {\left (x \right )}}{3} + \frac {25}{x} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx=-\frac {2}{3} \, x + \frac {25}{x} + 2 \, e^{\left (x \log \left (x\right ) - x\right )} - \frac {1}{3} \, \log \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx=-\frac {2 \, x^{2} - 6 \, x e^{\left (x \log \left (x\right ) - x\right )} + x \log \left (x\right ) - 75}{3 \, x} \]

[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

Mupad [B] (verification not implemented)

Time = 8.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-75-x-2 x^2+6 e^{-x+x \log (x)} x^2 \log (x)}{3 x^2} \, dx=\frac {25}{x}-\frac {\ln \left (x\right )}{3}-\frac {2\,x}{3}+2\,x^x\,{\mathrm {e}}^{-x} \]

[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)