3.38.50 \(\int \frac {e^{-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}} (3 x^2-6 x^3+3 x^4+e^{9+x} (6 x-3 x^2))}{5-10 x+5 x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac {1}{5} e^{\frac {3 e^{e^{-9-x} x} x^2}{1-x}} \]

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Rubi [F]  time = 8.29, 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 {\exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) \left (3 x^2-6 x^3+3 x^4+e^{9+x} \left (6 x-3 x^2\right )\right )}{5-10 x+5 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-9 - x + E^(-9 - x)*x - (3*E^(E^(-9 - x)*x)*x^2)/(-1 + x))*(3*x^2 - 6*x^3 + 3*x^4 + E^(9 + x)*(6*x - 3
*x^2)))/(5 - 10*x + 5*x^2),x]

[Out]

(-3*Defer[Int][E^(E^(-9 - x)*x - (3*E^(E^(-9 - x)*x)*x^2)/(-1 + x)), x])/5 + (3*Defer[Int][E^(E^(-9 - x)*x - (
3*E^(E^(-9 - x)*x)*x^2)/(-1 + x))/(-1 + x)^2, x])/5 + (3*Defer[Int][E^(-9 - x + E^(-9 - x)*x - (3*E^(E^(-9 - x
)*x)*x^2)/(-1 + x))*x^2, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) \left (3 x^2-6 x^3+3 x^4+e^{9+x} \left (6 x-3 x^2\right )\right )}{5 (-1+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) \left (3 x^2-6 x^3+3 x^4+e^{9+x} \left (6 x-3 x^2\right )\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {3 \exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) x \left (2 e^{9+x}+x-e^{9+x} x-2 x^2+x^3\right )}{(1-x)^2} \, dx\\ &=\frac {3}{5} \int \frac {\exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) x \left (2 e^{9+x}+x-e^{9+x} x-2 x^2+x^3\right )}{(1-x)^2} \, dx\\ &=\frac {3}{5} \int \left (-\frac {e^{e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}} (-2+x) x}{(-1+x)^2}+\exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) x^2\right ) \, dx\\ &=-\left (\frac {3}{5} \int \frac {e^{e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}} (-2+x) x}{(-1+x)^2} \, dx\right )+\frac {3}{5} \int \exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) x^2 \, dx\\ &=-\left (\frac {3}{5} \int \left (e^{e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}}-\frac {e^{e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}}}{(-1+x)^2}\right ) \, dx\right )+\frac {3}{5} \int \exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) x^2 \, dx\\ &=-\left (\frac {3}{5} \int e^{e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}} \, dx\right )+\frac {3}{5} \int \frac {e^{e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}}}{(-1+x)^2} \, dx+\frac {3}{5} \int \exp \left (-9-x+e^{-9-x} x-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}\right ) x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.84, size = 27, normalized size = 0.93 \begin {gather*} \frac {1}{5} e^{-\frac {3 e^{e^{-9-x} x} x^2}{-1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-9 - x + E^(-9 - x)*x - (3*E^(E^(-9 - x)*x)*x^2)/(-1 + x))*(3*x^2 - 6*x^3 + 3*x^4 + E^(9 + x)*(6
*x - 3*x^2)))/(5 - 10*x + 5*x^2),x]

[Out]

1/(5*E^((3*E^(E^(-9 - x)*x)*x^2)/(-1 + x)))

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fricas [B]  time = 1.07, size = 65, normalized size = 2.24 \begin {gather*} \frac {1}{5} \, e^{\left (-x e^{\left (-x - 9\right )} + x - \frac {{\left (3 \, x^{2} e^{\left (x e^{\left (-x - 9\right )} + x + 9\right )} - x^{2} + {\left (x^{2} + 8 \, x - 9\right )} e^{\left (x + 9\right )} + x\right )} e^{\left (-x - 9\right )}}{x - 1} + 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+6*x)*exp(x+9)+3*x^4-6*x^3+3*x^2)*exp(x/exp(x+9))*exp(-3*x^2*exp(x/exp(x+9))/(x-1))/(5*x^2-1
0*x+5)/exp(x+9),x, algorithm="fricas")

[Out]

1/5*e^(-x*e^(-x - 9) + x - (3*x^2*e^(x*e^(-x - 9) + x + 9) - x^2 + (x^2 + 8*x - 9)*e^(x + 9) + x)*e^(-x - 9)/(
x - 1) + 9)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, {\left (x^{4} - 2 \, x^{3} + x^{2} - {\left (x^{2} - 2 \, x\right )} e^{\left (x + 9\right )}\right )} e^{\left (-\frac {3 \, x^{2} e^{\left (x e^{\left (-x - 9\right )}\right )}}{x - 1} + x e^{\left (-x - 9\right )} - x - 9\right )}}{5 \, {\left (x^{2} - 2 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+6*x)*exp(x+9)+3*x^4-6*x^3+3*x^2)*exp(x/exp(x+9))*exp(-3*x^2*exp(x/exp(x+9))/(x-1))/(5*x^2-1
0*x+5)/exp(x+9),x, algorithm="giac")

[Out]

integrate(3/5*(x^4 - 2*x^3 + x^2 - (x^2 - 2*x)*e^(x + 9))*e^(-3*x^2*e^(x*e^(-x - 9))/(x - 1) + x*e^(-x - 9) -
x - 9)/(x^2 - 2*x + 1), x)

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maple [A]  time = 0.14, size = 23, normalized size = 0.79

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {3 x^{2} {\mathrm e}^{x \,{\mathrm e}^{-x -9}}}{x -1}}}{5}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^2+6*x)*exp(x+9)+3*x^4-6*x^3+3*x^2)*exp(x/exp(x+9))*exp(-3*x^2*exp(x/exp(x+9))/(x-1))/(5*x^2-10*x+5)
/exp(x+9),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.55, size = 43, normalized size = 1.48 \begin {gather*} \frac {1}{5} \, e^{\left (-3 \, x e^{\left (x e^{\left (-x - 9\right )}\right )} - \frac {3 \, e^{\left (x e^{\left (-x - 9\right )}\right )}}{x - 1} - 3 \, e^{\left (x e^{\left (-x - 9\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+6*x)*exp(x+9)+3*x^4-6*x^3+3*x^2)*exp(x/exp(x+9))*exp(-3*x^2*exp(x/exp(x+9))/(x-1))/(5*x^2-1
0*x+5)/exp(x+9),x, algorithm="maxima")

[Out]

1/5*e^(-3*x*e^(x*e^(-x - 9)) - 3*e^(x*e^(-x - 9))/(x - 1) - 3*e^(x*e^(-x - 9)))

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mupad [B]  time = 2.38, size = 22, normalized size = 0.76 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {3\,x^2\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-9}}}{x-1}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- x - 9)*exp(-(3*x^2*exp(x*exp(- x - 9)))/(x - 1))*exp(x*exp(- x - 9))*(exp(x + 9)*(6*x - 3*x^2) + 3*
x^2 - 6*x^3 + 3*x^4))/(5*x^2 - 10*x + 5),x)

[Out]

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

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sympy [A]  time = 0.66, size = 22, normalized size = 0.76 \begin {gather*} \frac {e^{- \frac {3 x^{2} e^{x e^{- x - 9}}}{x - 1}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**2+6*x)*exp(x+9)+3*x**4-6*x**3+3*x**2)*exp(x/exp(x+9))*exp(-3*x**2*exp(x/exp(x+9))/(x-1))/(5*
x**2-10*x+5)/exp(x+9),x)

[Out]

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

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