3.94.70 \(\int \frac {1}{3} e^{\frac {1}{16} (-9-24 e^5-16 e^{10})} x^{\frac {2}{3} e^{\frac {1}{16} (-9-24 e^5-16 e^{10})} x} (2+2 \log (x)) \, dx\)

Optimal. Leaf size=20 \[ x^{\frac {2}{3} e^{-\left (\frac {3}{4}+e^5\right )^2} x} \]

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Rubi [F]  time = 0.95, 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 {1}{3} e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x^{\frac {2}{3} e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} (2+2 \log (x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-9 - 24*E^5 - 16*E^10)/16)*x^((2*E^((-9 - 24*E^5 - 16*E^10)/16)*x)/3)*(2 + 2*Log[x]))/3,x]

[Out]

(2*Defer[Subst][Defer[Int][3^(2*E^((-9 - 24*E^5 - 16*E^10)/16)*x)*x^(2*E^((-9 - 24*E^5 - 16*E^10)/16)*x), x],
x, x/3])/E^((3 + 4*E^5)^2/16) + (2*Log[x]*Defer[Subst][Defer[Int][3^(2*E^((-9 - 24*E^5 - 16*E^10)/16)*x)*x^(2*
E^((-9 - 24*E^5 - 16*E^10)/16)*x), x], x, x/3])/E^((3 + 4*E^5)^2/16) - (2*Defer[Subst][Defer[Int][Defer[Int][3
^((2*x)/E^((3 + 4*E^5)^2/16))*x^((2*x)/E^((3 + 4*E^5)^2/16)), x]/x, x], x, x/3])/E^((3 + 4*E^5)^2/16)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} e^{-\frac {1}{16} \left (3+4 e^5\right )^2} \int x^{\frac {2}{3} e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} (2+2 \log (x)) \, dx\\ &=e^{-\frac {1}{16} \left (3+4 e^5\right )^2} \operatorname {Subst}\left (\int 3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} (2+2 \log (3 x)) \, dx,x,\frac {x}{3}\right )\\ &=e^{-\frac {1}{16} \left (3+4 e^5\right )^2} \operatorname {Subst}\left (\int 2\ 3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} (1+\log (3 x)) \, dx,x,\frac {x}{3}\right )\\ &=\left (2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2}\right ) \operatorname {Subst}\left (\int 3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} (1+\log (3 x)) \, dx,x,\frac {x}{3}\right )\\ &=\left (2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2}\right ) \operatorname {Subst}\left (\int \left (3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x}+3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} \log (3 x)\right ) \, dx,x,\frac {x}{3}\right )\\ &=\left (2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2}\right ) \operatorname {Subst}\left (\int 3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} \, dx,x,\frac {x}{3}\right )+\left (2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2}\right ) \operatorname {Subst}\left (\int 3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} \log (3 x) \, dx,x,\frac {x}{3}\right )\\ &=\left (2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2}\right ) \operatorname {Subst}\left (\int 3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} \, dx,x,\frac {x}{3}\right )-\left (2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2}\right ) \operatorname {Subst}\left (\int \frac {\int 3^{2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2} x} x^{2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2} x} \, dx}{x} \, dx,x,\frac {x}{3}\right )+\left (2 e^{-\frac {1}{16} \left (3+4 e^5\right )^2} \log (x)\right ) \operatorname {Subst}\left (\int 3^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} x^{2 e^{\frac {1}{16} \left (-9-24 e^5-16 e^{10}\right )} x} \, dx,x,\frac {x}{3}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 22, normalized size = 1.10 \begin {gather*} x^{\frac {2}{3} e^{-\frac {1}{16} \left (3+4 e^5\right )^2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-9 - 24*E^5 - 16*E^10)/16)*x^((2*E^((-9 - 24*E^5 - 16*E^10)/16)*x)/3)*(2 + 2*Log[x]))/3,x]

[Out]

x^((2*x)/(3*E^((3 + 4*E^5)^2/16)))

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fricas [A]  time = 0.59, size = 16, normalized size = 0.80 \begin {gather*} x^{\frac {2}{3} \, x e^{\left (-e^{10} - \frac {3}{2} \, e^{5} - \frac {9}{16}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*log(x)+2)*exp(1/3*x*log(x)/exp(exp(5)^2+3/2*exp(5)+9/16))^2/exp(exp(5)^2+3/2*exp(5)+9/16),x,
algorithm="fricas")

[Out]

x^(2/3*x*e^(-e^10 - 3/2*e^5 - 9/16))

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giac [A]  time = 0.91, size = 16, normalized size = 0.80 \begin {gather*} x^{\frac {2}{3} \, x e^{\left (-e^{10} - \frac {3}{2} \, e^{5} - \frac {9}{16}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*log(x)+2)*exp(1/3*x*log(x)/exp(exp(5)^2+3/2*exp(5)+9/16))^2/exp(exp(5)^2+3/2*exp(5)+9/16),x,
algorithm="giac")

[Out]

x^(2/3*x*e^(-e^10 - 3/2*e^5 - 9/16))

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maple [A]  time = 0.24, size = 19, normalized size = 0.95

method	result	size
risch	\(x^{\frac {2 x \,{\mathrm e}^{-{\mathrm e}^{10}-\frac {3 \,{\mathrm e}^{5}}{2}-\frac {9}{16}}}{3}}\)	\(19\)
norman	\({\mathrm e}^{x \,{\mathrm e}^{-{\mathrm e}^{10}-\frac {3 \,{\mathrm e}^{5}}{2}-\frac {9}{16}} \ln \left (x^{\frac {2}{3}}\right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(2*ln(x)+2)*exp(1/3*x*ln(x)/exp(exp(5)^2+3/2*exp(5)+9/16))^2/exp(exp(5)^2+3/2*exp(5)+9/16),x,method=_R
ETURNVERBOSE)

[Out]

(x^(1/3*x*exp(-exp(10)-3/2*exp(5)-9/16)))^2

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maxima [A]  time = 0.41, size = 16, normalized size = 0.80 \begin {gather*} x^{\frac {2}{3} \, x e^{\left (-e^{10} - \frac {3}{2} \, e^{5} - \frac {9}{16}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*log(x)+2)*exp(1/3*x*log(x)/exp(exp(5)^2+3/2*exp(5)+9/16))^2/exp(exp(5)^2+3/2*exp(5)+9/16),x,
algorithm="maxima")

[Out]

x^(2/3*x*e^(-e^10 - 3/2*e^5 - 9/16))

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mupad [B]  time = 7.46, size = 16, normalized size = 0.80 \begin {gather*} x^{\frac {2\,x\,{\mathrm {e}}^{-\frac {3\,{\mathrm {e}}^5}{2}-{\mathrm {e}}^{10}-\frac {9}{16}}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- (3*exp(5))/2 - exp(10) - 9/16)*exp((2*x*exp(- (3*exp(5))/2 - exp(10) - 9/16)*log(x))/3)*(2*log(x) +
2))/3,x)

[Out]

x^((2*x*exp(- (3*exp(5))/2 - exp(10) - 9/16))/3)

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sympy [A]  time = 0.33, size = 24, normalized size = 1.20 \begin {gather*} e^{\frac {2 x \log {\relax (x )}}{3 e^{\frac {9}{16} + \frac {3 e^{5}}{2} + e^{10}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*ln(x)+2)*exp(1/3*x*ln(x)/exp(exp(5)**2+3/2*exp(5)+9/16))**2/exp(exp(5)**2+3/2*exp(5)+9/16),x)

[Out]

exp(2*x*exp(-exp(10) - 3*exp(5)/2 - 9/16)*log(x)/3)

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