3.90.32 \(\int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} (8 x+6 x^2+e^5 x^2+2 x^3)}{4+4 x+x^2} \, dx\)

Optimal. Leaf size=20 \[ e^{e^{-\frac {e^5+x}{2+x}} x^2} \]

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Rubi [F]  time = 2.99, 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^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \left (8 x+6 x^2+e^5 x^2+2 x^3\right )}{4+4 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x^2/E^((E^5 + x)/(2 + x)) - (E^5 + x)/(2 + x))*(8*x + 6*x^2 + E^5*x^2 + 2*x^3))/(4 + 4*x + x^2),x]

[Out]

4*Defer[Int][E^(x^2/E^((E^5 + x)/(2 + x)) - (E^5 + x)/(2 + x)), x] - (6 - E^5)*Defer[Int][E^(x^2/E^((E^5 + x)/
(2 + x)) - (E^5 + x)/(2 + x)), x] + 2*Defer[Int][E^(x^2/E^((E^5 + x)/(2 + x)) - (E^5 + x)/(2 + x))*x, x] - 4*(
2 - E^5)*Defer[Int][E^(x^2/E^((E^5 + x)/(2 + x)) - (E^5 + x)/(2 + x))/(2 + x)^2, x] + 4*(2 - E^5)*Defer[Int][E
^(x^2/E^((E^5 + x)/(2 + x)) - (E^5 + x)/(2 + x))/(2 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \left (8 x+\left (6+e^5\right ) x^2+2 x^3\right )}{4+4 x+x^2} \, dx\\ &=\int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \left (8 x+\left (6+e^5\right ) x^2+2 x^3\right )}{(2+x)^2} \, dx\\ &=\int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} x \left (8+\left (6+e^5\right ) x+2 x^2\right )}{(2+x)^2} \, dx\\ &=\int \left (e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \left (-6+e^5\right )+\frac {4 e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \left (-2+e^5\right )}{(2+x)^2}-\frac {4 e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \left (-2+e^5\right )}{2+x}+2 e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} (2+x)\right ) \, dx\\ &=2 \int e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} (2+x) \, dx-\left (4 \left (2-e^5\right )\right ) \int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}}}{(2+x)^2} \, dx+\left (4 \left (2-e^5\right )\right ) \int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}}}{2+x} \, dx+\left (-6+e^5\right ) \int e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \, dx\\ &=2 \int \left (2 e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}}+e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} x\right ) \, dx-\left (4 \left (2-e^5\right )\right ) \int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}}}{(2+x)^2} \, dx+\left (4 \left (2-e^5\right )\right ) \int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}}}{2+x} \, dx+\left (-6+e^5\right ) \int e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \, dx\\ &=2 \int e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} x \, dx+4 \int e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \, dx-\left (4 \left (2-e^5\right )\right ) \int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}}}{(2+x)^2} \, dx+\left (4 \left (2-e^5\right )\right ) \int \frac {e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}}}{2+x} \, dx+\left (-6+e^5\right ) \int e^{e^{-\frac {e^5+x}{2+x}} x^2-\frac {e^5+x}{2+x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.75, size = 20, normalized size = 1.00 \begin {gather*} e^{e^{-\frac {e^5+x}{2+x}} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2/E^((E^5 + x)/(2 + x)) - (E^5 + x)/(2 + x))*(8*x + 6*x^2 + E^5*x^2 + 2*x^3))/(4 + 4*x + x^2),
x]

[Out]

E^(x^2/E^((E^5 + x)/(2 + x)))

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fricas [B]  time = 0.53, size = 48, normalized size = 2.40 \begin {gather*} e^{\left (\frac {{\left (x^{3} + 2 \, x^{2}\right )} e^{\left (-\frac {x + e^{5}}{x + 2}\right )} - x - e^{5}}{x + 2} + \frac {x + e^{5}}{x + 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(5)+2*x^3+6*x^2+8*x)*exp(x^2/exp((exp(5)+x)/(2+x)))/(x^2+4*x+4)/exp((exp(5)+x)/(2+x)),x, alg
orithm="fricas")

[Out]

e^(((x^3 + 2*x^2)*e^(-(x + e^5)/(x + 2)) - x - e^5)/(x + 2) + (x + e^5)/(x + 2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{3} + x^{2} e^{5} + 6 \, x^{2} + 8 \, x\right )} e^{\left (x^{2} e^{\left (-\frac {x + e^{5}}{x + 2}\right )} - \frac {x + e^{5}}{x + 2}\right )}}{x^{2} + 4 \, x + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(5)+2*x^3+6*x^2+8*x)*exp(x^2/exp((exp(5)+x)/(2+x)))/(x^2+4*x+4)/exp((exp(5)+x)/(2+x)),x, alg
orithm="giac")

[Out]

integrate((2*x^3 + x^2*e^5 + 6*x^2 + 8*x)*e^(x^2*e^(-(x + e^5)/(x + 2)) - (x + e^5)/(x + 2))/(x^2 + 4*x + 4),
x)

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maple [A]  time = 0.46, size = 18, normalized size = 0.90

method	result	size
risch	\({\mathrm e}^{x^{2} {\mathrm e}^{-\frac {{\mathrm e}^{5}+x}{2+x}}}\)	\(18\)
norman	\(\frac {\left (x \,{\mathrm e}^{\frac {{\mathrm e}^{5}+x}{2+x}} {\mathrm e}^{x^{2} {\mathrm e}^{-\frac {{\mathrm e}^{5}+x}{2+x}}}+2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}+x}{2+x}} {\mathrm e}^{x^{2} {\mathrm e}^{-\frac {{\mathrm e}^{5}+x}{2+x}}}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{5}+x}{2+x}}}{2+x}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(5)+2*x^3+6*x^2+8*x)*exp(x^2/exp((exp(5)+x)/(2+x)))/(x^2+4*x+4)/exp((exp(5)+x)/(2+x)),x,method=_RE
TURNVERBOSE)

[Out]

exp(x^2*exp(-(exp(5)+x)/(2+x)))

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maxima [A]  time = 0.68, size = 24, normalized size = 1.20 \begin {gather*} e^{\left (x^{2} e^{\left (-\frac {e^{5}}{x + 2} + \frac {2}{x + 2} - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(5)+2*x^3+6*x^2+8*x)*exp(x^2/exp((exp(5)+x)/(2+x)))/(x^2+4*x+4)/exp((exp(5)+x)/(2+x)),x, alg
orithm="maxima")

[Out]

e^(x^2*e^(-e^5/(x + 2) + 2/(x + 2) - 1))

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mupad [B]  time = 8.89, size = 17, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{x^2\,{\mathrm {e}}^{-\frac {x+{\mathrm {e}}^5}{x+2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x + exp(5))/(x + 2))*exp(x^2*exp(-(x + exp(5))/(x + 2)))*(8*x + x^2*exp(5) + 6*x^2 + 2*x^3))/(4*x +
x^2 + 4),x)

[Out]

exp(x^2*exp(-(x + exp(5))/(x + 2)))

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sympy [A]  time = 0.61, size = 14, normalized size = 0.70 \begin {gather*} e^{x^{2} e^{- \frac {x + e^{5}}{x + 2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(5)+2*x**3+6*x**2+8*x)*exp(x**2/exp((exp(5)+x)/(2+x)))/(x**2+4*x+4)/exp((exp(5)+x)/(2+x)),x
)

[Out]

exp(x**2*exp(-(x + exp(5))/(x + 2)))

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