3.6.54 \(\int \frac {1}{3} (3+2 x+e^{-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4} (48 x-144 x^2+96 x^3)) \, dx\)

Optimal. Leaf size=31 \[ e^{2 e^{4 \left (-2+e^3+\left (-x+x^2\right )^2\right )}}+x+\frac {x^2}{3} \]

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Rubi [F]  time = 1.60, 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 {1}{3} \left (3+2 x+\exp \left (-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4\right ) \left (48 x-144 x^2+96 x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 + 2*x + E^(-8 + 4*E^3 + 2*E^(-8 + 4*E^3 + 4*x^2 - 8*x^3 + 4*x^4) + 4*x^2 - 8*x^3 + 4*x^4)*(48*x - 144*x
^2 + 96*x^3))/3,x]

[Out]

x + x^2/3 + 16*Defer[Int][E^(2*E^(-8 + 4*E^3 + 4*x^2 - 8*x^3 + 4*x^4) - 8*(1 - E^3/2) + 4*x^2 - 8*x^3 + 4*x^4)
*x, x] - 48*Defer[Int][E^(2*E^(-8 + 4*E^3 + 4*x^2 - 8*x^3 + 4*x^4) - 8*(1 - E^3/2) + 4*x^2 - 8*x^3 + 4*x^4)*x^
2, x] + 32*Defer[Int][E^(2*E^(-8 + 4*E^3 + 4*x^2 - 8*x^3 + 4*x^4) - 8*(1 - E^3/2) + 4*x^2 - 8*x^3 + 4*x^4)*x^3
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (3+2 x+\exp \left (-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4\right ) \left (48 x-144 x^2+96 x^3\right )\right ) \, dx\\ &=x+\frac {x^2}{3}+\frac {1}{3} \int \exp \left (-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4\right ) \left (48 x-144 x^2+96 x^3\right ) \, dx\\ &=x+\frac {x^2}{3}+\frac {1}{3} \int \exp \left (-8+4 e^3+2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}+4 x^2-8 x^3+4 x^4\right ) x \left (48-144 x+96 x^2\right ) \, dx\\ &=x+\frac {x^2}{3}+\frac {1}{3} \int \exp \left (2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}-8 \left (1-\frac {e^3}{2}\right )+4 x^2-8 x^3+4 x^4\right ) x \left (48-144 x+96 x^2\right ) \, dx\\ &=x+\frac {x^2}{3}+\frac {1}{3} \int \left (48 \exp \left (2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}-8 \left (1-\frac {e^3}{2}\right )+4 x^2-8 x^3+4 x^4\right ) x-144 \exp \left (2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}-8 \left (1-\frac {e^3}{2}\right )+4 x^2-8 x^3+4 x^4\right ) x^2+96 \exp \left (2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}-8 \left (1-\frac {e^3}{2}\right )+4 x^2-8 x^3+4 x^4\right ) x^3\right ) \, dx\\ &=x+\frac {x^2}{3}+16 \int \exp \left (2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}-8 \left (1-\frac {e^3}{2}\right )+4 x^2-8 x^3+4 x^4\right ) x \, dx+32 \int \exp \left (2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}-8 \left (1-\frac {e^3}{2}\right )+4 x^2-8 x^3+4 x^4\right ) x^3 \, dx-48 \int \exp \left (2 e^{-8+4 e^3+4 x^2-8 x^3+4 x^4}-8 \left (1-\frac {e^3}{2}\right )+4 x^2-8 x^3+4 x^4\right ) x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.54, size = 38, normalized size = 1.23 \begin {gather*} e^{2 e^{4 \left (-2+e^3\right )+4 x^2-8 x^3+4 x^4}}+x+\frac {x^2}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x + E^(-8 + 4*E^3 + 2*E^(-8 + 4*E^3 + 4*x^2 - 8*x^3 + 4*x^4) + 4*x^2 - 8*x^3 + 4*x^4)*(48*x -
144*x^2 + 96*x^3))/3,x]

[Out]

E^(2*E^(4*(-2 + E^3) + 4*x^2 - 8*x^3 + 4*x^4)) + x + x^2/3

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fricas [B]  time = 0.71, size = 103, normalized size = 3.32 \begin {gather*} \frac {1}{3} \, {\left ({\left (x^{2} + 3 \, x\right )} e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )} + 3 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} + 2 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )} - 8\right )}\right )} e^{\left (-4 \, x^{4} + 8 \, x^{3} - 4 \, x^{2} - 4 \, e^{3} + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp(exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2
+2/3*x+1,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 16 \, {\left (2 \, x^{3} - 3 \, x^{2} + x\right )} e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} + 2 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )} - 8\right )} + \frac {2}{3} \, x + 1\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp(exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2
+2/3*x+1,x, algorithm="giac")

[Out]

integrate(16*(2*x^3 - 3*x^2 + x)*e^(4*x^4 - 8*x^3 + 4*x^2 + 4*e^3 + 2*e^(4*x^4 - 8*x^3 + 4*x^2 + 4*e^3 - 8) -
8) + 2/3*x + 1, x)

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maple [A]  time = 0.22, size = 33, normalized size = 1.06

method	result	size
default	\(x +\frac {x^{2}}{3}+{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}\)	\(33\)
norman	\(x +\frac {x^{2}}{3}+{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}\)	\(33\)
risch	\(x +\frac {x^{2}}{3}+{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{3}+4 x^{4}-8 x^{3}+4 x^{2}-8}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp(exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2+2/3*x
+1,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 32, normalized size = 1.03 \begin {gather*} \frac {1}{3} \, x^{2} + x + e^{\left (2 \, e^{\left (4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2} + 4 \, e^{3} - 8\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(96*x^3-144*x^2+48*x)*exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8)*exp(exp(4*exp(3)+4*x^4-8*x^3+4*x^2-8))^2
+2/3*x+1,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.60, size = 35, normalized size = 1.13 \begin {gather*} x+{\mathrm {e}}^{2\,{\mathrm {e}}^{4\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-8}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{4\,x^4}\,{\mathrm {e}}^{-8\,x^3}}+\frac {x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x)/3 + (exp(4*exp(3) + 4*x^2 - 8*x^3 + 4*x^4 - 8)*exp(2*exp(4*exp(3) + 4*x^2 - 8*x^3 + 4*x^4 - 8))*(48*
x - 144*x^2 + 96*x^3))/3 + 1,x)

[Out]

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

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sympy [A]  time = 0.37, size = 32, normalized size = 1.03 \begin {gather*} \frac {x^{2}}{3} + x + e^{2 e^{4 x^{4} - 8 x^{3} + 4 x^{2} - 8 + 4 e^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(96*x**3-144*x**2+48*x)*exp(4*exp(3)+4*x**4-8*x**3+4*x**2-8)*exp(exp(4*exp(3)+4*x**4-8*x**3+4*x*
*2-8))**2+2/3*x+1,x)

[Out]

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

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