3.72.87 \(\int \frac {e^{-\frac {5 x^2+x^3-\log (5)-x^2 \log (x)}{x^2}} (2 x^2-x^3-2 \log (5))}{x^2} \, dx\)

Optimal. Leaf size=19 \[ 2+e^{-5-x+\frac {\log (5)}{x^2}} x^2 \]

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Rubi [A]  time = 0.33, antiderivative size = 25, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6688, 2288} \begin {gather*} \frac {5^{\frac {1}{x^2}} e^{-x-5} x^2 \log (25)}{2 \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^2 - x^3 - 2*Log[5])/(E^((5*x^2 + x^3 - Log[5] - x^2*Log[x])/x^2)*x^2),x]

[Out]

(5^x^(-2)*E^(-5 - x)*x^2*Log[25])/(2*Log[5])

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5^{\frac {1}{x^2}} e^{-5-x} \left (2 x^2-x^3-\log (25)\right )}{x} \, dx\\ &=\frac {5^{\frac {1}{x^2}} e^{-5-x} x^2 \log (25)}{2 \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.84 \begin {gather*} 5^{\frac {1}{x^2}} e^{-5-x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 - x^3 - 2*Log[5])/(E^((5*x^2 + x^3 - Log[5] - x^2*Log[x])/x^2)*x^2),x]

[Out]

5^x^(-2)*E^(-5 - x)*x^2

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fricas [A]  time = 0.50, size = 28, normalized size = 1.47 \begin {gather*} x e^{\left (-\frac {x^{3} - x^{2} \log \relax (x) + 5 \, x^{2} - \log \relax (5)}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(5)-x^3+2*x^2)/x^2/exp((-x^2*log(x)-log(5)+x^3+5*x^2)/x^2),x, algorithm="fricas")

[Out]

x*e^(-(x^3 - x^2*log(x) + 5*x^2 - log(5))/x^2)

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giac [A]  time = 0.16, size = 23, normalized size = 1.21 \begin {gather*} x^{2} e^{\left (-\frac {x^{3} + 5 \, x^{2} - \log \relax (5)}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(5)-x^3+2*x^2)/x^2/exp((-x^2*log(x)-log(5)+x^3+5*x^2)/x^2),x, algorithm="giac")

[Out]

x^2*e^(-(x^3 + 5*x^2 - log(5))/x^2)

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maple [A]  time = 0.07, size = 18, normalized size = 0.95

method	result	size
risch	\(x^{2} 5^{\frac {1}{x^{2}}} {\mathrm e}^{-x -5}\)	\(18\)
gosper	\(x \,{\mathrm e}^{\frac {x^{2} \ln \relax (x )-x^{3}-5 x^{2}+\ln \relax (5)}{x^{2}}}\)	\(30\)
norman	\(x \,{\mathrm e}^{-\frac {-x^{2} \ln \relax (x )-\ln \relax (5)+x^{3}+5 x^{2}}{x^{2}}}\)	\(30\)
meijerg	\(-2 \,5^{1+\frac {1}{x^{2}}} {\mathrm e}^{4 x -5} x \ln \relax (5) \left (-\frac {1}{5 x}+1-\ln \relax (5)-\ln \relax (x )+\frac {2-10 x}{10 x}-\frac {{\mathrm e}^{-5 x}}{5 x}+\ln \left (5 x \right )+\expIntegralEi \left (1, 5 x \right )\right )-5^{-2+\frac {1}{x^{2}}} {\mathrm e}^{4 x -5} x \left (1-\frac {\left (10 x +2\right ) {\mathrm e}^{-5 x}}{2}\right )+2 \,5^{-1+\frac {1}{x^{2}}} {\mathrm e}^{4 x -5} x \left (1-{\mathrm e}^{-5 x}\right )\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(5)-x^3+2*x^2)/x^2/exp((-x^2*ln(x)-ln(5)+x^3+5*x^2)/x^2),x,method=_RETURNVERBOSE)

[Out]

x^2/((1/5)^(1/x^2))*exp(-x-5)

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maxima [A]  time = 0.52, size = 16, normalized size = 0.84 \begin {gather*} x^{2} e^{\left (-x + \frac {\log \relax (5)}{x^{2}} - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(5)-x^3+2*x^2)/x^2/exp((-x^2*log(x)-log(5)+x^3+5*x^2)/x^2),x, algorithm="maxima")

[Out]

x^2*e^(-x + log(5)/x^2 - 5)

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mupad [B]  time = 4.59, size = 15, normalized size = 0.79 \begin {gather*} 5^{\frac {1}{x^2}}\,x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((log(5) + x^2*log(x) - 5*x^2 - x^3)/x^2)*(2*log(5) - 2*x^2 + x^3))/x^2,x)

[Out]

5^(1/x^2)*x^2*exp(-x)*exp(-5)

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sympy [A]  time = 0.35, size = 24, normalized size = 1.26 \begin {gather*} x e^{- \frac {x^{3} - x^{2} \log {\relax (x )} + 5 x^{2} - \log {\relax (5 )}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(5)-x**3+2*x**2)/x**2/exp((-x**2*ln(x)-ln(5)+x**3+5*x**2)/x**2),x)

[Out]

x*exp(-(x**3 - x**2*log(x) + 5*x**2 - log(5))/x**2)

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