3.80.94 \(\int \frac {1}{2} e^{-x-\frac {1}{2} e^{-x} (-675 x+27 x^2)} (4 e^x x+675 x^2-729 x^3+27 x^4) \, dx\)

Optimal. Leaf size=19 \[ e^{-\frac {27}{2} e^{-x} (-25+x) x} x^2 \]

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Rubi [F]  time = 0.93, 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}{2} e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x - (-675*x + 27*x^2)/(2*E^x))*(4*E^x*x + 675*x^2 - 729*x^3 + 27*x^4))/2,x]

[Out]

2*Defer[Int][x/E^((27*(-25 + x)*x)/(2*E^x)), x] + (675*Defer[Int][x^2/E^((x*(-675 + 2*E^x + 27*x))/(2*E^x)), x
])/2 - (729*Defer[Int][x^3/E^((x*(-675 + 2*E^x + 27*x))/(2*E^x)), x])/2 + (27*Defer[Int][x^4/E^((x*(-675 + 2*E
^x + 27*x))/(2*E^x)), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{-x-\frac {1}{2} e^{-x} \left (-675 x+27 x^2\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx\\ &=\frac {1}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} \left (4 e^x x+675 x^2-729 x^3+27 x^4\right ) \, dx\\ &=\frac {1}{2} \int \left (4 e^{x-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x+675 e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^2-729 e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^3+27 e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^4\right ) \, dx\\ &=2 \int e^{x-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x \, dx+\frac {27}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^4 \, dx+\frac {675}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^2 \, dx-\frac {729}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^3 \, dx\\ &=2 \int e^{-\frac {27}{2} e^{-x} (-25+x) x} x \, dx+\frac {27}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^4 \, dx+\frac {675}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^2 \, dx-\frac {729}{2} \int e^{-\frac {1}{2} e^{-x} x \left (-675+2 e^x+27 x\right )} x^3 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 19, normalized size = 1.00 \begin {gather*} e^{-\frac {27}{2} e^{-x} (-25+x) x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x - (-675*x + 27*x^2)/(2*E^x))*(4*E^x*x + 675*x^2 - 729*x^3 + 27*x^4))/2,x]

[Out]

x^2/E^((27*(-25 + x)*x)/(2*E^x))

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fricas [A]  time = 0.69, size = 27, normalized size = 1.42 \begin {gather*} x^{2} e^{\left (-\frac {1}{2} \, {\left (27 \, x^{2} + 2 \, x e^{x} - 675 \, x\right )} e^{\left (-x\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-675*x)/exp(x)),x, algorithm="fricas")

[Out]

x^2*e^(-1/2*(27*x^2 + 2*x*e^x - 675*x)*e^(-x) + x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{2} \, {\left (27 \, x^{4} - 729 \, x^{3} + 675 \, x^{2} + 4 \, x e^{x}\right )} e^{\left (-\frac {27}{2} \, {\left (x^{2} - 25 \, x\right )} e^{\left (-x\right )} - x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-675*x)/exp(x)),x, algorithm="giac")

[Out]

integrate(1/2*(27*x^4 - 729*x^3 + 675*x^2 + 4*x*e^x)*e^(-27/2*(x^2 - 25*x)*e^(-x) - x), x)

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maple [A]  time = 0.07, size = 16, normalized size = 0.84

method	result	size
risch	\(x^{2} {\mathrm e}^{-\frac {27 x \left (x -25\right ) {\mathrm e}^{-x}}{2}}\)	\(16\)
norman	\(x^{2} {\mathrm e}^{-\frac {\left (27 x^{2}-675 x \right ) {\mathrm e}^{-x}}{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-675*x)/exp(x)),x,method=_RETURNVERBOSE)

[Out]

x^2*exp(-27/2*x*(x-25)*exp(-x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{2} \, \int {\left (27 \, x^{4} - 729 \, x^{3} + 675 \, x^{2} + 4 \, x e^{x}\right )} e^{\left (-\frac {27}{2} \, {\left (x^{2} - 25 \, x\right )} e^{\left (-x\right )} - x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(x)*x+27*x^4-729*x^3+675*x^2)/exp(x)/exp(1/2*(27*x^2-675*x)/exp(x)),x, algorithm="maxima")

[Out]

1/2*integrate((27*x^4 - 729*x^3 + 675*x^2 + 4*x*e^x)*e^(-27/2*(x^2 - 25*x)*e^(-x) - x), x)

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mupad [B]  time = 6.52, size = 22, normalized size = 1.16 \begin {gather*} x^2\,{\mathrm {e}}^{\frac {675\,x\,{\mathrm {e}}^{-x}}{2}}\,{\mathrm {e}}^{-\frac {27\,x^2\,{\mathrm {e}}^{-x}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*exp(exp(-x)*((675*x)/2 - (27*x^2)/2))*(2*x*exp(x) + (675*x^2)/2 - (729*x^3)/2 + (27*x^4)/2),x)

[Out]

x^2*exp((675*x*exp(-x))/2)*exp(-(27*x^2*exp(-x))/2)

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sympy [A]  time = 0.31, size = 19, normalized size = 1.00 \begin {gather*} x^{2} e^{- \left (\frac {27 x^{2}}{2} - \frac {675 x}{2}\right ) e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*exp(x)*x+27*x**4-729*x**3+675*x**2)/exp(x)/exp(1/2*(27*x**2-675*x)/exp(x)),x)

[Out]

x**2*exp(-(27*x**2/2 - 675*x/2)*exp(-x))

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