3.67.78 \(\int \frac {e^{\frac {20+x}{x}} (-40-2 x)}{x^3} \, dx\)

Optimal. Leaf size=24 \[ \frac {2 \left (e^{\frac {20+x}{x}}+\left (3-e^{32}\right ) x\right )}{x} \]

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Rubi [A]  time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2288} \begin {gather*} -\frac {40 e^{\frac {x+20}{x}}}{x^3 \left (\frac {1}{x}-\frac {x+20}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((20 + x)/x)*(-40 - 2*x))/x^3,x]

[Out]

(-40*E^((20 + x)/x))/(x^3*(x^(-1) - (20 + x)/x^2))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {40 e^{\frac {20+x}{x}}}{x^3 \left (\frac {1}{x}-\frac {20+x}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.58 \begin {gather*} \frac {2 e^{1+\frac {20}{x}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((20 + x)/x)*(-40 - 2*x))/x^3,x]

[Out]

(2*E^(1 + 20/x))/x

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fricas [A]  time = 0.46, size = 13, normalized size = 0.54 \begin {gather*} \frac {2 \, e^{\left (\frac {x + 20}{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x^3,x, algorithm="fricas")

[Out]

2*e^((x + 20)/x)/x

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giac [A]  time = 0.19, size = 27, normalized size = 1.12 \begin {gather*} \frac {{\left (x + 20\right )} e^{\left (\frac {x + 20}{x}\right )}}{10 \, x} - \frac {1}{10} \, e^{\left (\frac {x + 20}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x^3,x, algorithm="giac")

[Out]

1/10*(x + 20)*e^((x + 20)/x)/x - 1/10*e^((x + 20)/x)

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maple [A]  time = 0.06, size = 14, normalized size = 0.58

method	result	size
gosper	\(\frac {2 \,{\mathrm e}^{\frac {20+x}{x}}}{x}\)	\(14\)
norman	\(\frac {2 \,{\mathrm e}^{\frac {20+x}{x}}}{x}\)	\(14\)
risch	\(\frac {2 \,{\mathrm e}^{\frac {20+x}{x}}}{x}\)	\(14\)
derivativedivides	\(\frac {\left (1+\frac {20}{x}\right ) {\mathrm e}^{1+\frac {20}{x}}}{10}-\frac {{\mathrm e}^{1+\frac {20}{x}}}{10}\)	\(29\)
default	\(\frac {\left (1+\frac {20}{x}\right ) {\mathrm e}^{1+\frac {20}{x}}}{10}-\frac {{\mathrm e}^{1+\frac {20}{x}}}{10}\)	\(29\)
meijerg	\(-\frac {{\mathrm e} \left (1-{\mathrm e}^{\frac {20}{x}}\right )}{10}+\frac {{\mathrm e} \left (1-\frac {\left (2-\frac {40}{x}\right ) {\mathrm e}^{\frac {20}{x}}}{2}\right )}{10}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-40)*exp((20+x)/x)/x^3,x,method=_RETURNVERBOSE)

[Out]

2*exp((20+x)/x)/x

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maxima [C]  time = 0.46, size = 22, normalized size = 0.92 \begin {gather*} -\frac {1}{10} \, e \Gamma \left (2, -\frac {20}{x}\right ) + \frac {1}{10} \, e^{\left (\frac {20}{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x^3,x, algorithm="maxima")

[Out]

-1/10*e*gamma(2, -20/x) + 1/10*e^(20/x + 1)

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mupad [B]  time = 4.12, size = 13, normalized size = 0.54 \begin {gather*} \frac {2\,\mathrm {e}\,{\mathrm {e}}^{20/x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x + 20)/x)*(2*x + 40))/x^3,x)

[Out]

(2*exp(1)*exp(20/x))/x

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sympy [A]  time = 0.11, size = 8, normalized size = 0.33 \begin {gather*} \frac {2 e^{\frac {x + 20}{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-40)*exp((20+x)/x)/x**3,x)

[Out]

2*exp((x + 20)/x)/x

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