3.93.38 \(\int \frac {1}{81} (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} (405+4096 e^{\frac {256 x^4}{81}} x^3)) \, dx\)

Optimal. Leaf size=23 \[ 2+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x}+2 x \]

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Rubi [A]  time = 0.09, antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {12, 6706} \begin {gather*} e^{4 e^{\frac {256 x^4}{81}}+5 x-1}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(162 + E^(-1 + 4*E^((256*x^4)/81) + 5*x)*(405 + 4096*E^((256*x^4)/81)*x^3))/81,x]

[Out]

E^(-1 + 4*E^((256*x^4)/81) + 5*x) + 2*x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \left (162+e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right )\right ) \, dx\\ &=2 x+\frac {1}{81} \int e^{-1+4 e^{\frac {256 x^4}{81}}+5 x} \left (405+4096 e^{\frac {256 x^4}{81}} x^3\right ) \, dx\\ &=e^{-1+4 e^{\frac {256 x^4}{81}}+5 x}+2 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 22, normalized size = 0.96 \begin {gather*} e^{-1+4 e^{\frac {256 x^4}{81}}+5 x}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(162 + E^(-1 + 4*E^((256*x^4)/81) + 5*x)*(405 + 4096*E^((256*x^4)/81)*x^3))/81,x]

[Out]

E^(-1 + 4*E^((256*x^4)/81) + 5*x) + 2*x

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fricas [A]  time = 1.03, size = 18, normalized size = 0.78 \begin {gather*} 2 \, x + e^{\left (5 \, x + 4 \, e^{\left (\frac {256}{81} \, x^{4}\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x, algorithm="fricas")

[Out]

2*x + e^(5*x + 4*e^(256/81*x^4) - 1)

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giac [A]  time = 0.16, size = 18, normalized size = 0.78 \begin {gather*} 2 \, x + e^{\left (5 \, x + 4 \, e^{\left (\frac {256}{81} \, x^{4}\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x, algorithm="giac")

[Out]

2*x + e^(5*x + 4*e^(256/81*x^4) - 1)

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maple [A]  time = 0.06, size = 19, normalized size = 0.83

method	result	size
default	\(2 x +{\mathrm e}^{4 \,{\mathrm e}^{\frac {256 x^{4}}{81}}+5 x -1}\)	\(19\)
norman	\(2 x +{\mathrm e}^{4 \,{\mathrm e}^{\frac {256 x^{4}}{81}}+5 x -1}\)	\(19\)
risch	\(2 x +{\mathrm e}^{4 \,{\mathrm e}^{\frac {256 x^{4}}{81}}+5 x -1}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x,method=_RETURNVERBOSE)

[Out]

2*x+exp(4*exp(256/81*x^4)+5*x-1)

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maxima [A]  time = 0.36, size = 18, normalized size = 0.78 \begin {gather*} 2 \, x + e^{\left (5 \, x + 4 \, e^{\left (\frac {256}{81} \, x^{4}\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(4096*x^3*exp(256/81*x^4)+405)*exp(4*exp(256/81*x^4)+5*x-1)+2,x, algorithm="maxima")

[Out]

2*x + e^(5*x + 4*e^(256/81*x^4) - 1)

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mupad [B]  time = 6.98, size = 23, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{-1}\,\left (2\,x\,\mathrm {e}+{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{\frac {256\,x^4}{81}}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5*x + 4*exp((256*x^4)/81) - 1)*(4096*x^3*exp((256*x^4)/81) + 405))/81 + 2,x)

[Out]

exp(-1)*(2*x*exp(1) + exp(5*x)*exp(4*exp((256*x^4)/81)))

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sympy [A]  time = 0.48, size = 19, normalized size = 0.83 \begin {gather*} 2 x + e^{5 x + 4 e^{\frac {256 x^{4}}{81}} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(4096*x**3*exp(256/81*x**4)+405)*exp(4*exp(256/81*x**4)+5*x-1)+2,x)

[Out]

2*x + exp(5*x + 4*exp(256*x**4/81) - 1)

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