3.86.31 \(\int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{2 x+2 x^2}} (4+12 x+9 x^2+6 x^3+x^4)}{4+8 x+4 x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{2} e^{\frac {1}{2} \left (-9+x-\frac {4}{x+x^2}\right )} x^2 \]

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Rubi [F]  time = 1.05, 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^{\frac {-4-9 x-8 x^2+x^3}{2 x+2 x^2}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{4+8 x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-4 - 9*x - 8*x^2 + x^3)/(2*x + 2*x^2))*(4 + 12*x + 9*x^2 + 6*x^3 + x^4))/(4 + 8*x + 4*x^2),x]

[Out]

Defer[Int][E^((-4 - 9*x - 8*x^2 + x^3)/(x*(2 + 2*x)))*x, x] + Defer[Int][E^((-4 - 9*x - 8*x^2 + x^3)/(x*(2 + 2
*x)))*x^2, x]/4 - Defer[Int][E^((-4 - 9*x - 8*x^2 + x^3)/(x*(2 + 2*x)))/(1 + x)^2, x] + 2*Defer[Int][E^((-4 -
9*x - 8*x^2 + x^3)/(x*(2 + 2*x)))/(1 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{2 x+2 x^2}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{4 (1+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{2 x+2 x^2}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{(1+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{(1+x)^2} \, dx\\ &=\frac {1}{4} \int \left (4 e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x+e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x^2-\frac {4 e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{(1+x)^2}+\frac {8 e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{1+x}\right ) \, dx\\ &=\frac {1}{4} \int e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x^2 \, dx+2 \int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{1+x} \, dx+\int e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x \, dx-\int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{(1+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 30, normalized size = 1.20 \begin {gather*} \frac {1}{2} e^{-\frac {9}{2}-\frac {2}{x}+\frac {x}{2}+\frac {2}{1+x}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-4 - 9*x - 8*x^2 + x^3)/(2*x + 2*x^2))*(4 + 12*x + 9*x^2 + 6*x^3 + x^4))/(4 + 8*x + 4*x^2),x]

[Out]

(E^(-9/2 - 2/x + x/2 + 2/(1 + x))*x^2)/2

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fricas [A]  time = 1.24, size = 28, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (\frac {x^{3} - 8 \, x^{2} - 9 \, x - 4}{2 \, {\left (x^{2} + x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6*x^3+9*x^2+12*x+4)*exp((x^3-8*x^2-9*x-4)/(2*x^2+2*x))/(4*x^2+8*x+4),x, algorithm="fricas")

[Out]

1/2*x^2*e^(1/2*(x^3 - 8*x^2 - 9*x - 4)/(x^2 + x))

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giac [A]  time = 0.22, size = 28, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (\frac {x^{3} - 8 \, x^{2} - 9 \, x - 4}{2 \, {\left (x^{2} + x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6*x^3+9*x^2+12*x+4)*exp((x^3-8*x^2-9*x-4)/(2*x^2+2*x))/(4*x^2+8*x+4),x, algorithm="giac")

[Out]

1/2*x^2*e^(1/2*(x^3 - 8*x^2 - 9*x - 4)/(x^2 + x))

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maple [A]  time = 0.41, size = 30, normalized size = 1.20

method	result	size
gosper	\(\frac {x^{2} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 \left (x +1\right ) x}}}{2}\)	\(30\)
risch	\(\frac {x^{2} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 \left (x +1\right ) x}}}{2}\)	\(30\)
norman	\(\frac {\frac {x^{2} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 x^{2}+2 x}}}{2}+\frac {x^{3} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 x^{2}+2 x}}}{2}}{x +1}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+6*x^3+9*x^2+12*x+4)*exp((x^3-8*x^2-9*x-4)/(2*x^2+2*x))/(4*x^2+8*x+4),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2*exp(1/2*(x^3-8*x^2-9*x-4)/(x+1)/x)

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maxima [A]  time = 0.47, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (\frac {1}{2} \, x + \frac {2}{x + 1} - \frac {2}{x} - \frac {9}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6*x^3+9*x^2+12*x+4)*exp((x^3-8*x^2-9*x-4)/(2*x^2+2*x))/(4*x^2+8*x+4),x, algorithm="maxima")

[Out]

1/2*x^2*e^(1/2*x + 2/(x + 1) - 2/x - 9/2)

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mupad [B]  time = 5.36, size = 46, normalized size = 1.84 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{\frac {x^2}{2\,x+2}}\,{\mathrm {e}}^{-\frac {9}{2\,x+2}}\,{\mathrm {e}}^{-\frac {4\,x}{x+1}}\,{\mathrm {e}}^{-\frac {2}{x^2+x}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(9*x + 8*x^2 - x^3 + 4)/(2*x + 2*x^2))*(12*x + 9*x^2 + 6*x^3 + x^4 + 4))/(8*x + 4*x^2 + 4),x)

[Out]

(x^2*exp(x^2/(2*x + 2))*exp(-9/(2*x + 2))*exp(-(4*x)/(x + 1))*exp(-2/(x + x^2)))/2

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sympy [A]  time = 0.26, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^{2} e^{\frac {x^{3} - 8 x^{2} - 9 x - 4}{2 x^{2} + 2 x}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+6*x**3+9*x**2+12*x+4)*exp((x**3-8*x**2-9*x-4)/(2*x**2+2*x))/(4*x**2+8*x+4),x)

[Out]

x**2*exp((x**3 - 8*x**2 - 9*x - 4)/(2*x**2 + 2*x))/2

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