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

Optimal. Leaf size=22 \[ -25+e^{\frac {5-x+e^{-x} x^2}{x}} \]

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Rubi [F]  time = 1.13, 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} \left (e^x (5-x)+x^2\right )}{x}} \left (-5 e^x+x^2-x^3\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.29, size = 16, normalized size = 0.73 \begin {gather*} e^{-1+\frac {5}{x}+e^{-x} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-1 + 5/x + x/E^x)

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fricas [A]  time = 0.63, size = 25, normalized size = 1.14 \begin {gather*} e^{\left (x + \frac {{\left (x^{2} - {\left (x^{2} + x - 5\right )} e^{x}\right )} e^{\left (-x\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.14, size = 22, normalized size = 1.00

method	result	size
norman	\({\mathrm e}^{\frac {\left (\left (5-x \right ) {\mathrm e}^{x}+x^{2}\right ) {\mathrm e}^{-x}}{x}}\)	\(22\)
risch	\({\mathrm e}^{-\frac {\left ({\mathrm e}^{x} x -x^{2}-5 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.57, size = 14, normalized size = 0.64 \begin {gather*} e^{\left (x e^{\left (-x\right )} + \frac {5}{x} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x*e^(-x) + 5/x - 1)

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mupad [B]  time = 1.31, size = 16, normalized size = 0.73 \begin {gather*} {\mathrm {e}}^{-1}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{5/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.21, size = 15, normalized size = 0.68 \begin {gather*} e^{\frac {\left (x^{2} + \left (5 - x\right ) e^{x}\right ) e^{- x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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