3.67.17 \(\int \frac {e^{-\frac {-9+6 x+3 x^2}{x}} (-9+x-3 x^2)}{x} \, dx\)

Optimal. Leaf size=16 \[ -5+e^{-3 \left (2-\frac {3}{x}+x\right )} x \]

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Rubi [B]  time = 0.07, antiderivative size = 51, normalized size of antiderivative = 3.19, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2288} \begin {gather*} \frac {e^{\frac {3 \left (-x^2-2 x+3\right )}{x}} \left (x^2+3\right )}{x \left (\frac {-x^2-2 x+3}{x^2}+\frac {2 (x+1)}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 + x - 3*x^2)/(E^((-9 + 6*x + 3*x^2)/x)*x),x]

[Out]

(E^((3*(3 - 2*x - x^2))/x)*(3 + x^2))/(x*((2*(1 + x))/x + (3 - 2*x - x^2)/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 {e^{\frac {3 \left (3-2 x-x^2\right )}{x}} \left (3+x^2\right )}{x \left (\frac {2 (1+x)}{x}+\frac {3-2 x-x^2}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.88 \begin {gather*} e^{-6+\frac {9}{x}-3 x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 + x - 3*x^2)/(E^((-9 + 6*x + 3*x^2)/x)*x),x]

[Out]

E^(-6 + 9/x - 3*x)*x

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fricas [A]  time = 0.56, size = 16, normalized size = 1.00 \begin {gather*} x e^{\left (-\frac {3 \, {\left (x^{2} + 2 \, x - 3\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+x-9)/x/exp((3*x^2+6*x-9)/x),x, algorithm="fricas")

[Out]

x*e^(-3*(x^2 + 2*x - 3)/x)

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giac [A]  time = 0.13, size = 16, normalized size = 1.00 \begin {gather*} x e^{\left (-\frac {3 \, {\left (x^{2} + 2 \, x - 3\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+x-9)/x/exp((3*x^2+6*x-9)/x),x, algorithm="giac")

[Out]

x*e^(-3*(x^2 + 2*x - 3)/x)

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maple [A]  time = 0.17, size = 15, normalized size = 0.94

method	result	size
risch	\(x \,{\mathrm e}^{-\frac {3 \left (3+x \right ) \left (x -1\right )}{x}}\)	\(15\)
gosper	\(x \,{\mathrm e}^{-\frac {3 \left (x^{2}+2 x -3\right )}{x}}\)	\(19\)
norman	\(x \,{\mathrm e}^{-\frac {3 x^{2}+6 x -9}{x}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^2+x-9)/x/exp((3*x^2+6*x-9)/x),x,method=_RETURNVERBOSE)

[Out]

x*exp(-3*(3+x)*(x-1)/x)

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maxima [A]  time = 0.51, size = 13, normalized size = 0.81 \begin {gather*} x e^{\left (-3 \, x + \frac {9}{x} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+x-9)/x/exp((3*x^2+6*x-9)/x),x, algorithm="maxima")

[Out]

x*e^(-3*x + 9/x - 6)

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mupad [B]  time = 4.26, size = 14, normalized size = 0.88 \begin {gather*} x\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{9/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(6*x + 3*x^2 - 9)/x)*(3*x^2 - x + 9))/x,x)

[Out]

x*exp(-3*x)*exp(-6)*exp(9/x)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.88 \begin {gather*} x e^{- \frac {3 x^{2} + 6 x - 9}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**2+x-9)/x/exp((3*x**2+6*x-9)/x),x)

[Out]

x*exp(-(3*x**2 + 6*x - 9)/x)

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