3.98.88 \(\int \frac {e^{-x-\frac {e^{-x} x}{5}} (-10 e^x+2 x-3 x^2+x^3)}{20-20 x+5 x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {e^{-\frac {1}{5} e^{-x} x} x}{-2+x} \]

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Rubi [F]  time = 0.92, 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 {e^{-x-\frac {e^{-x} x}{5}} \left (-10 e^x+2 x-3 x^2+x^3\right )}{20-20 x+5 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x - x/(5*E^x))*(-10*E^x + 2*x - 3*x^2 + x^3))/(20 - 20*x + 5*x^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 25, normalized size = 1.32 \begin {gather*} \frac {1}{5} e^{-\frac {1}{5} e^{-x} x} \left (5+\frac {10}{-2+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x - x/(5*E^x))*(-10*E^x + 2*x - 3*x^2 + x^3))/(20 - 20*x + 5*x^2),x]

[Out]

(5 + 10/(-2 + x))/(5*E^(x/(5*E^x)))

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fricas [A]  time = 0.64, size = 23, normalized size = 1.21 \begin {gather*} \frac {x e^{\left (-\frac {1}{5} \, {\left (5 \, x e^{x} + x\right )} e^{\left (-x\right )} + x\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(x)+x^3-3*x^2+2*x)/(5*x^2-20*x+20)/exp(x)/exp(1/5*x/exp(x)),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 3 \, x^{2} + 2 \, x - 10 \, e^{x}\right )} e^{\left (-\frac {1}{5} \, x e^{\left (-x\right )} - x\right )}}{5 \, {\left (x^{2} - 4 \, x + 4\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(x)+x^3-3*x^2+2*x)/(5*x^2-20*x+20)/exp(x)/exp(1/5*x/exp(x)),x, algorithm="giac")

[Out]

integrate(1/5*(x^3 - 3*x^2 + 2*x - 10*e^x)*e^(-1/5*x*e^(-x) - x)/(x^2 - 4*x + 4), x)

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maple [A]  time = 0.12, size = 16, normalized size = 0.84

method	result	size
risch	\(\frac {x \,{\mathrm e}^{-\frac {x \,{\mathrm e}^{-x}}{5}}}{x -2}\)	\(16\)
norman	\(\frac {x \,{\mathrm e}^{-\frac {x \,{\mathrm e}^{-x}}{5}}}{x -2}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*exp(x)+x^3-3*x^2+2*x)/(5*x^2-20*x+20)/exp(x)/exp(1/5*x/exp(x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{5} \, \int \frac {{\left (x^{3} - 3 \, x^{2} + 2 \, x - 10 \, e^{x}\right )} e^{\left (-\frac {1}{5} \, x e^{\left (-x\right )} - x\right )}}{x^{2} - 4 \, x + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(x)+x^3-3*x^2+2*x)/(5*x^2-20*x+20)/exp(x)/exp(1/5*x/exp(x)),x, algorithm="maxima")

[Out]

1/5*integrate((x^3 - 3*x^2 + 2*x - 10*e^x)*e^(-1/5*x*e^(-x) - x)/(x^2 - 4*x + 4), x)

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mupad [B]  time = 5.77, size = 21, normalized size = 1.11 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x-\frac {x\,{\mathrm {e}}^{-x}}{5}}}{x-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*exp(-(x*exp(-x))/5)*(2*x - 10*exp(x) - 3*x^2 + x^3))/(5*x^2 - 20*x + 20),x)

[Out]

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

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sympy [A]  time = 0.17, size = 12, normalized size = 0.63 \begin {gather*} \frac {x e^{- \frac {x e^{- x}}{5}}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(x)+x**3-3*x**2+2*x)/(5*x**2-20*x+20)/exp(x)/exp(1/5*x/exp(x)),x)

[Out]

x*exp(-x*exp(-x)/5)/(x - 2)

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