3.67.99 \(\int \frac {x-4 x^2+e^{\frac {-2+11 x-x^2}{x}} (2+x-x^2)}{x} \, dx\)

Optimal. Leaf size=19 \[ \left (1+e^{11-\frac {2}{x}-x}-2 x\right ) x \]

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Rubi [F]  time = 0.30, 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 {x-4 x^2+e^{\frac {-2+11 x-x^2}{x}} \left (2+x-x^2\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x - 4*x^2 + E^((-2 + 11*x - x^2)/x)*(2 + x - x^2))/x,x]

[Out]

x - 2*x^2 + Defer[Int][E^(11 - 2/x - x), x] + 2*Defer[Int][E^(11 - 2/x - x)/x, x] - Defer[Int][E^(11 - 2/x - x
)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-4 x-\frac {e^{11-\frac {2}{x}-x} (-2+x) (1+x)}{x}\right ) \, dx\\ &=x-2 x^2-\int \frac {e^{11-\frac {2}{x}-x} (-2+x) (1+x)}{x} \, dx\\ &=x-2 x^2-\int \left (-e^{11-\frac {2}{x}-x}-\frac {2 e^{11-\frac {2}{x}-x}}{x}+e^{11-\frac {2}{x}-x} x\right ) \, dx\\ &=x-2 x^2+2 \int \frac {e^{11-\frac {2}{x}-x}}{x} \, dx+\int e^{11-\frac {2}{x}-x} \, dx-\int e^{11-\frac {2}{x}-x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 19, normalized size = 1.00 \begin {gather*} \left (1+e^{11-\frac {2}{x}-x}-2 x\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x - 4*x^2 + E^((-2 + 11*x - x^2)/x)*(2 + x - x^2))/x,x]

[Out]

(1 + E^(11 - 2/x - x) - 2*x)*x

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fricas [A]  time = 0.54, size = 23, normalized size = 1.21 \begin {gather*} -2 \, x^{2} + x e^{\left (-\frac {x^{2} - 11 \, x + 2}{x}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+x+2)*exp((-x^2+11*x-2)/x)-4*x^2+x)/x,x, algorithm="fricas")

[Out]

-2*x^2 + x*e^(-(x^2 - 11*x + 2)/x) + x

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giac [A]  time = 0.13, size = 23, normalized size = 1.21 \begin {gather*} -2 \, x^{2} + x e^{\left (-\frac {x^{2} - 11 \, x + 2}{x}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+x+2)*exp((-x^2+11*x-2)/x)-4*x^2+x)/x,x, algorithm="giac")

[Out]

-2*x^2 + x*e^(-(x^2 - 11*x + 2)/x) + x

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maple [A]  time = 0.16, size = 24, normalized size = 1.26

method	result	size
risch	\(x +{\mathrm e}^{-\frac {x^{2}-11 x +2}{x}} x -2 x^{2}\)	\(24\)
norman	\(x +x \,{\mathrm e}^{\frac {-x^{2}+11 x -2}{x}}-2 x^{2}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+x+2)*exp((-x^2+11*x-2)/x)-4*x^2+x)/x,x,method=_RETURNVERBOSE)

[Out]

x+exp(-(x^2-11*x+2)/x)*x-2*x^2

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maxima [A]  time = 0.43, size = 20, normalized size = 1.05 \begin {gather*} -2 \, x^{2} + x e^{\left (-x - \frac {2}{x} + 11\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+x+2)*exp((-x^2+11*x-2)/x)-4*x^2+x)/x,x, algorithm="maxima")

[Out]

-2*x^2 + x*e^(-x - 2/x + 11) + x

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mupad [B]  time = 4.12, size = 21, normalized size = 1.11 \begin {gather*} x-2\,x^2+x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{11}\,{\mathrm {e}}^{-\frac {2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(-(x^2 - 11*x + 2)/x)*(x - x^2 + 2) - 4*x^2)/x,x)

[Out]

x - 2*x^2 + x*exp(-x)*exp(11)*exp(-2/x)

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sympy [A]  time = 0.13, size = 19, normalized size = 1.00 \begin {gather*} - 2 x^{2} + x e^{\frac {- x^{2} + 11 x - 2}{x}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+x+2)*exp((-x**2+11*x-2)/x)-4*x**2+x)/x,x)

[Out]

-2*x**2 + x*exp((-x**2 + 11*x - 2)/x) + x

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