3.58.38 \(\int \frac {e^{-2 x} (e^{3+2 x} x^2+e^{3+\frac {12 e^{-2 x}}{x}} (12+24 x))}{x^2} \, dx\)

Optimal. Leaf size=21 \[ e^3 \left (2-e^{\frac {12 e^{-2 x}}{x}}+x\right ) \]

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Rubi [F]  time = 0.69, 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^{-2 x} \left (e^{3+2 x} x^2+e^{3+\frac {12 e^{-2 x}}{x}} (12+24 x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3 + 2*x)*x^2 + E^(3 + 12/(E^(2*x)*x))*(12 + 24*x))/(E^(2*x)*x^2),x]

[Out]

E^3*x + 12*Defer[Int][E^(3 + 12/(E^(2*x)*x) - 2*x)/x^2, x] + 24*Defer[Int][E^(3 + 12/(E^(2*x)*x) - 2*x)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^3+\frac {12 e^{3+\frac {12 e^{-2 x}}{x}-2 x} (1+2 x)}{x^2}\right ) \, dx\\ &=e^3 x+12 \int \frac {e^{3+\frac {12 e^{-2 x}}{x}-2 x} (1+2 x)}{x^2} \, dx\\ &=e^3 x+12 \int \left (\frac {e^{3+\frac {12 e^{-2 x}}{x}-2 x}}{x^2}+\frac {2 e^{3+\frac {12 e^{-2 x}}{x}-2 x}}{x}\right ) \, dx\\ &=e^3 x+12 \int \frac {e^{3+\frac {12 e^{-2 x}}{x}-2 x}}{x^2} \, dx+24 \int \frac {e^{3+\frac {12 e^{-2 x}}{x}-2 x}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 22, normalized size = 1.05 \begin {gather*} -e^{3+\frac {12 e^{-2 x}}{x}}+e^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 + 2*x)*x^2 + E^(3 + 12/(E^(2*x)*x))*(12 + 24*x))/(E^(2*x)*x^2),x]

[Out]

-E^(3 + 12/(E^(2*x)*x)) + E^3*x

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fricas [A]  time = 0.79, size = 32, normalized size = 1.52 \begin {gather*} x e^{3} - e^{\left (\frac {3 \, {\left (x e^{\left (2 \, x + 3\right )} + 4 \, e^{3}\right )} e^{\left (-2 \, x - 3\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+12)*exp(3)*exp(12/x/exp(2*x))+x^2*exp(3)*exp(2*x))/exp(2*x)/x^2,x, algorithm="fricas")

[Out]

x*e^3 - e^(3*(x*e^(2*x + 3) + 4*e^3)*e^(-2*x - 3)/x)

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giac [B]  time = 0.18, size = 37, normalized size = 1.76 \begin {gather*} {\left (x e^{\left (-2 \, x + 3\right )} - e^{\left (-\frac {2 \, x^{2} - 3 \, x - 12 \, e^{\left (-2 \, x\right )}}{x}\right )}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+12)*exp(3)*exp(12/x/exp(2*x))+x^2*exp(3)*exp(2*x))/exp(2*x)/x^2,x, algorithm="giac")

[Out]

(x*e^(-2*x + 3) - e^(-(2*x^2 - 3*x - 12*e^(-2*x))/x))*e^(2*x)

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maple [A]  time = 0.07, size = 22, normalized size = 1.05

method	result	size
risch	\(x \,{\mathrm e}^{3}-{\mathrm e}^{\frac {3 x +12 \,{\mathrm e}^{-2 x}}{x}}\)	\(22\)
norman	\(\frac {\left (x^{2} {\mathrm e}^{3} {\mathrm e}^{2 x}-x \,{\mathrm e}^{3} {\mathrm e}^{2 x} {\mathrm e}^{\frac {12 \,{\mathrm e}^{-2 x}}{x}}\right ) {\mathrm e}^{-2 x}}{x}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x+12)*exp(3)*exp(12/x/exp(2*x))+x^2*exp(3)*exp(2*x))/exp(2*x)/x^2,x,method=_RETURNVERBOSE)

[Out]

x*exp(3)-exp(3*(x+4*exp(-2*x))/x)

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maxima [A]  time = 0.54, size = 19, normalized size = 0.90 \begin {gather*} x e^{3} - e^{\left (\frac {12 \, e^{\left (-2 \, x\right )}}{x} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+12)*exp(3)*exp(12/x/exp(2*x))+x^2*exp(3)*exp(2*x))/exp(2*x)/x^2,x, algorithm="maxima")

[Out]

x*e^3 - e^(12*e^(-2*x)/x + 3)

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mupad [B]  time = 3.60, size = 17, normalized size = 0.81 \begin {gather*} {\mathrm {e}}^3\,\left (x-{\mathrm {e}}^{\frac {12\,{\mathrm {e}}^{-2\,x}}{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*x)*(exp(3)*exp((12*exp(-2*x))/x)*(24*x + 12) + x^2*exp(2*x)*exp(3)))/x^2,x)

[Out]

exp(3)*(x - exp((12*exp(-2*x))/x))

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sympy [A]  time = 0.20, size = 17, normalized size = 0.81 \begin {gather*} x e^{3} - e^{3} e^{\frac {12 e^{- 2 x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+12)*exp(3)*exp(12/x/exp(2*x))+x**2*exp(3)*exp(2*x))/exp(2*x)/x**2,x)

[Out]

x*exp(3) - exp(3)*exp(12*exp(-2*x)/x)

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