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

Optimal. Leaf size=28 \[ e^{2 e^{e^{5+x^2}-x} x}+\left (-2+e^x\right ) x^3 \]

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Rubi [F]  time = 1.08, 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 \left (-6 x^2+e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \left (2-2 x+4 e^{5+x^2} x^2\right )+e^x \left (3 x^2+x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-2 x^3+\int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \left (2-2 x+4 e^{5+x^2} x^2\right ) \, dx+\int e^x \left (3 x^2+x^3\right ) \, dx\\ &=-2 x^3+\int e^x x^2 (3+x) \, dx+\int 2 e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \left (1-x+2 e^{5+x^2} x^2\right ) \, dx\\ &=-2 x^3+2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \left (1-x+2 e^{5+x^2} x^2\right ) \, dx+\int \left (3 e^x x^2+e^x x^3\right ) \, dx\\ &=-2 x^3+2 \int \left (e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x}-e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} x+2 e^{5+e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x+x^2} x^2\right ) \, dx+3 \int e^x x^2 \, dx+\int e^x x^3 \, dx\\ &=3 e^x x^2-2 x^3+e^x x^3+2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \, dx-2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} x \, dx-3 \int e^x x^2 \, dx+4 \int e^{5+e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x+x^2} x^2 \, dx-6 \int e^x x \, dx\\ &=-6 e^x x-2 x^3+e^x x^3+2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \, dx-2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} x \, dx+4 \int e^{5+e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x+x^2} x^2 \, dx+6 \int e^x \, dx+6 \int e^x x \, dx\\ &=6 e^x-2 x^3+e^x x^3+2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \, dx-2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} x \, dx+4 \int e^{5+e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x+x^2} x^2 \, dx-6 \int e^x \, dx\\ &=-2 x^3+e^x x^3+2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} \, dx-2 \int e^{e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x} x \, dx+4 \int e^{5+e^{5+x^2}-x+2 e^{e^{5+x^2}-x} x+x^2} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.72, size = 31, normalized size = 1.11 \begin {gather*} e^{2 e^{e^{5+x^2}-x} x}-2 x^3+e^x x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*E^(E^(5 + x^2) - x)*x) - 2*x^3 + E^x*x^3

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fricas [B]  time = 0.76, size = 62, normalized size = 2.21 \begin {gather*} {\left ({\left (x^{3} e^{x} - 2 \, x^{3}\right )} e^{\left (-x + e^{\left (x^{2} + 5\right )}\right )} + e^{\left (2 \, x e^{\left (-x + e^{\left (x^{2} + 5\right )}\right )} - x + e^{\left (x^{2} + 5\right )}\right )}\right )} e^{\left (x - e^{\left (x^{2} + 5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(5)*exp(x^2)-2*x+2)*exp(exp(5)*exp(x^2)-x)*exp(2*x*exp(exp(5)*exp(x^2)-x))+(x^3+3*x^2)*exp
(x)-6*x^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -6 \, x^{2} + 2 \, {\left (2 \, x^{2} e^{\left (x^{2} + 5\right )} - x + 1\right )} e^{\left (2 \, x e^{\left (-x + e^{\left (x^{2} + 5\right )}\right )} - x + e^{\left (x^{2} + 5\right )}\right )} + {\left (x^{3} + 3 \, x^{2}\right )} e^{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(5)*exp(x^2)-2*x+2)*exp(exp(5)*exp(x^2)-x)*exp(2*x*exp(exp(5)*exp(x^2)-x))+(x^3+3*x^2)*exp
(x)-6*x^2,x, algorithm="giac")

[Out]

integrate(-6*x^2 + 2*(2*x^2*e^(x^2 + 5) - x + 1)*e^(2*x*e^(-x + e^(x^2 + 5)) - x + e^(x^2 + 5)) + (x^3 + 3*x^2
)*e^x, x)

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maple [A]  time = 0.14, size = 28, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x \,{\mathrm e}^{{\mathrm e}^{x^{2}+5}-x}}-2 x^{3}\)	\(28\)
default	\({\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x \,{\mathrm e}^{{\mathrm e}^{5} {\mathrm e}^{x^{2}}-x}}-2 x^{3}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*exp(5)*exp(x^2)-2*x+2)*exp(exp(5)*exp(x^2)-x)*exp(2*x*exp(exp(5)*exp(x^2)-x))+(x^3+3*x^2)*exp(x)-6*
x^2,x,method=_RETURNVERBOSE)

[Out]

exp(x)*x^3+exp(2*x*exp(exp(x^2+5)-x))-2*x^3

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maxima [B]  time = 0.38, size = 49, normalized size = 1.75 \begin {gather*} -2 \, x^{3} + {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + 3 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + e^{\left (2 \, x e^{\left (-x + e^{\left (x^{2} + 5\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(5)*exp(x^2)-2*x+2)*exp(exp(5)*exp(x^2)-x)*exp(2*x*exp(exp(5)*exp(x^2)-x))+(x^3+3*x^2)*exp
(x)-6*x^2,x, algorithm="maxima")

[Out]

-2*x^3 + (x^3 - 3*x^2 + 6*x - 6)*e^x + 3*(x^2 - 2*x + 2)*e^x + e^(2*x*e^(-x + e^(x^2 + 5)))

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mupad [B]  time = 3.68, size = 28, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^5}}+x^3\,{\mathrm {e}}^x-2\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*x*exp(-x)*exp(exp(x^2)*exp(5))) + x^3*exp(x) - 2*x^3

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sympy [A]  time = 1.72, size = 27, normalized size = 0.96 \begin {gather*} x^{3} e^{x} - 2 x^{3} + e^{2 x e^{- x + e^{5} e^{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*exp(x) - 2*x**3 + exp(2*x*exp(-x + exp(5)*exp(x**2)))

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