3.99.89 \(\int \frac {e^{-x} (e^{2 e^{-x} x} (2500 x^2-2500 x^3)+e^x (1250 e^5+300 x^2+4 x^3)+e^{e^{-x} x} (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4))}{625 x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac {2 \left (-e^5+\left (-3+e^{e^{-x} x}-\frac {x}{25}\right )^2 x\right )}{x} \]

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Rubi [F]  time = 1.45, 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} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x)/E^x)*(2500*x^2 - 2500*x^3) + E^x*(1250*E^5 + 300*x^2 + 4*x^3) + E^(x/E^x)*(-7500*x^2 - 100*E^x*x
^2 + 7400*x^3 + 100*x^4))/(625*E^x*x^2),x]

[Out]

(-2*E^5)/x + (12*x)/25 + (2*x^2)/625 - (4*Defer[Int][E^(x/E^x), x])/25 + 4*Defer[Int][E^(-(((-2 + E^x)*x)/E^x)
), x] - 12*Defer[Int][E^(-(((-1 + E^x)*x)/E^x)), x] - 4*Defer[Int][x/E^(((-2 + E^x)*x)/E^x), x] + (296*Defer[I
nt][x/E^(((-1 + E^x)*x)/E^x), x])/25 + (4*Defer[Int][x^2/E^(((-1 + E^x)*x)/E^x), x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{x^2} \, dx\\ &=\frac {1}{625} \int \left (-2500 e^{-x+2 e^{-x} x} (-1+x)-100 e^{-x+e^{-x} x} \left (75+e^x-74 x-x^2\right )+\frac {2 \left (625 e^5+150 x^2+2 x^3\right )}{x^2}\right ) \, dx\\ &=\frac {2}{625} \int \frac {625 e^5+150 x^2+2 x^3}{x^2} \, dx-\frac {4}{25} \int e^{-x+e^{-x} x} \left (75+e^x-74 x-x^2\right ) \, dx-4 \int e^{-x+2 e^{-x} x} (-1+x) \, dx\\ &=\frac {2}{625} \int \left (150+\frac {625 e^5}{x^2}+2 x\right ) \, dx-\frac {4}{25} \int e^{-e^{-x} \left (-1+e^x\right ) x} \left (75+e^x-74 x-x^2\right ) \, dx-4 \int e^{-e^{-x} \left (-2+e^x\right ) x} (-1+x) \, dx\\ &=-\frac {2 e^5}{x}+\frac {12 x}{25}+\frac {2 x^2}{625}-\frac {4}{25} \int \left (75 e^{-e^{-x} \left (-1+e^x\right ) x}+e^{x-e^{-x} \left (-1+e^x\right ) x}-74 e^{-e^{-x} \left (-1+e^x\right ) x} x-e^{-e^{-x} \left (-1+e^x\right ) x} x^2\right ) \, dx-4 \int \left (-e^{-e^{-x} \left (-2+e^x\right ) x}+e^{-e^{-x} \left (-2+e^x\right ) x} x\right ) \, dx\\ &=-\frac {2 e^5}{x}+\frac {12 x}{25}+\frac {2 x^2}{625}-\frac {4}{25} \int e^{x-e^{-x} \left (-1+e^x\right ) x} \, dx+\frac {4}{25} \int e^{-e^{-x} \left (-1+e^x\right ) x} x^2 \, dx+4 \int e^{-e^{-x} \left (-2+e^x\right ) x} \, dx-4 \int e^{-e^{-x} \left (-2+e^x\right ) x} x \, dx+\frac {296}{25} \int e^{-e^{-x} \left (-1+e^x\right ) x} x \, dx-12 \int e^{-e^{-x} \left (-1+e^x\right ) x} \, dx\\ &=-\frac {2 e^5}{x}+\frac {12 x}{25}+\frac {2 x^2}{625}-\frac {4}{25} \int e^{e^{-x} x} \, dx+\frac {4}{25} \int e^{-e^{-x} \left (-1+e^x\right ) x} x^2 \, dx+4 \int e^{-e^{-x} \left (-2+e^x\right ) x} \, dx-4 \int e^{-e^{-x} \left (-2+e^x\right ) x} x \, dx+\frac {296}{25} \int e^{-e^{-x} \left (-1+e^x\right ) x} x \, dx-12 \int e^{-e^{-x} \left (-1+e^x\right ) x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 45, normalized size = 1.45 \begin {gather*} 2 e^{2 e^{-x} x}-\frac {2 e^5}{x}-\frac {4}{25} e^{e^{-x} x} (75+x)+\frac {2}{625} x (150+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*x)/E^x)*(2500*x^2 - 2500*x^3) + E^x*(1250*E^5 + 300*x^2 + 4*x^3) + E^(x/E^x)*(-7500*x^2 - 100
*E^x*x^2 + 7400*x^3 + 100*x^4))/(625*E^x*x^2),x]

[Out]

2*E^((2*x)/E^x) - (2*E^5)/x - (4*E^(x/E^x)*(75 + x))/25 + (2*x*(150 + x))/625

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fricas [A]  time = 0.59, size = 45, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left (x^{3} + 150 \, x^{2} + 625 \, x e^{\left (2 \, x e^{\left (-x\right )}\right )} - 50 \, {\left (x^{2} + 75 \, x\right )} e^{\left (x e^{\left (-x\right )}\right )} - 625 \, e^{5}\right )}}{625 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100*x^4+7400*x^3-7500*x^2)*exp(x/exp(x)
)+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/exp(x)/x^2,x, algorithm="fricas")

[Out]

2/625*(x^3 + 150*x^2 + 625*x*e^(2*x*e^(-x)) - 50*(x^2 + 75*x)*e^(x*e^(-x)) - 625*e^5)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (1250 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )} - 50 \, {\left (x^{4} + 74 \, x^{3} - x^{2} e^{x} - 75 \, x^{2}\right )} e^{\left (x e^{\left (-x\right )}\right )} - {\left (2 \, x^{3} + 150 \, x^{2} + 625 \, e^{5}\right )} e^{x}\right )} e^{\left (-x\right )}}{625 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100*x^4+7400*x^3-7500*x^2)*exp(x/exp(x)
)+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/exp(x)/x^2,x, algorithm="giac")

[Out]

integrate(-2/625*(1250*(x^3 - x^2)*e^(2*x*e^(-x)) - 50*(x^4 + 74*x^3 - x^2*e^x - 75*x^2)*e^(x*e^(-x)) - (2*x^3
+ 150*x^2 + 625*e^5)*e^x)*e^(-x)/x^2, x)

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maple [A]  time = 0.08, size = 41, normalized size = 1.32

method	result	size
risch	\(\frac {2 x^{2}}{625}+\frac {12 x}{25}-\frac {2 \,{\mathrm e}^{5}}{x}+2 \,{\mathrm e}^{2 x \,{\mathrm e}^{-x}}+\frac {\left (-100 x -7500\right ) {\mathrm e}^{x \,{\mathrm e}^{-x}}}{625}\)	\(41\)
norman	\(\frac {\left (-2 \,{\mathrm e}^{5} {\mathrm e}^{x}+\frac {12 \,{\mathrm e}^{x} x^{2}}{25}+\frac {2 \,{\mathrm e}^{x} x^{3}}{625}-12 \,{\mathrm e}^{x} x \,{\mathrm e}^{x \,{\mathrm e}^{-x}}+2 \,{\mathrm e}^{x} x \,{\mathrm e}^{2 x \,{\mathrm e}^{-x}}-\frac {4 \,{\mathrm e}^{x} {\mathrm e}^{x \,{\mathrm e}^{-x}} x^{2}}{25}\right ) {\mathrm e}^{-x}}{x}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100*x^4+7400*x^3-7500*x^2)*exp(x/exp(x))+(125
0*exp(5)+4*x^3+300*x^2)*exp(x))/exp(x)/x^2,x,method=_RETURNVERBOSE)

[Out]

2/625*x^2+12/25*x-2*exp(5)/x+2*exp(2*x*exp(-x))+1/625*(-100*x-7500)*exp(x*exp(-x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2}{625} \, x^{2} + \frac {12}{25} \, x - \frac {2 \, e^{5}}{x} + 2 \, e^{\left (2 \, x e^{\left (-x\right )}\right )} - \frac {2}{625} \, \int -50 \, {\left (x^{2} + 74 \, x - e^{x} - 75\right )} e^{\left (x e^{\left (-x\right )} - x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100*x^4+7400*x^3-7500*x^2)*exp(x/exp(x)
)+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/exp(x)/x^2,x, algorithm="maxima")

[Out]

2/625*x^2 + 12/25*x - 2*e^5/x + 2*e^(2*x*e^(-x)) - 2/625*integrate(-50*(x^2 + 74*x - e^x - 75)*e^(x*e^(-x) - x
), x)

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mupad [B]  time = 5.80, size = 41, normalized size = 1.32 \begin {gather*} 2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}+\frac {2\,\left (x^3+150\,x^2-625\,{\mathrm {e}}^5\right )}{625\,x}-\frac {4\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}\,\left (x+75\right )}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*((exp(2*x*exp(-x))*(2500*x^2 - 2500*x^3))/625 + (exp(x)*(1250*exp(5) + 300*x^2 + 4*x^3))/625 - (e
xp(x*exp(-x))*(100*x^2*exp(x) + 7500*x^2 - 7400*x^3 - 100*x^4))/625))/x^2,x)

[Out]

2*exp(2*x*exp(-x)) + (2*(150*x^2 - 625*exp(5) + x^3))/(625*x) - (4*exp(x*exp(-x))*(x + 75))/25

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sympy [A]  time = 0.34, size = 42, normalized size = 1.35 \begin {gather*} \frac {2 x^{2}}{625} + \frac {12 x}{25} + \frac {\left (- 4 x - 300\right ) e^{x e^{- x}}}{25} + 2 e^{2 x e^{- x}} - \frac {2 e^{5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((-2500*x**3+2500*x**2)*exp(x/exp(x))**2+(-100*exp(x)*x**2+100*x**4+7400*x**3-7500*x**2)*exp(x
/exp(x))+(1250*exp(5)+4*x**3+300*x**2)*exp(x))/exp(x)/x**2,x)

[Out]

2*x**2/625 + 12*x/25 + (-4*x - 300)*exp(x*exp(-x))/25 + 2*exp(2*x*exp(-x)) - 2*exp(5)/x

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