3.48.66 \(\int e^{104976-186624 x+124416 x^2-36864 x^3+4096 x^4} (-186624+248832 x-110592 x^2+16384 x^3) \, dx\)

Optimal. Leaf size=20 \[ e^{\left (2-\frac {3 (6-2 x)}{x}\right )^4 x^4} \]

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Rubi [A]  time = 0.62, antiderivative size = 11, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6688, 12, 2209} \begin {gather*} e^{16 (9-4 x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(104976 - 186624*x + 124416*x^2 - 36864*x^3 + 4096*x^4)*(-186624 + 248832*x - 110592*x^2 + 16384*x^3),x]

[Out]

E^(16*(9 - 4*x)^4)

Rule 12

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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 256 e^{16 (9-4 x)^4} (-9+4 x)^3 \, dx\\ &=256 \int e^{16 (9-4 x)^4} (-9+4 x)^3 \, dx\\ &=e^{16 (9-4 x)^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 11, normalized size = 0.55 \begin {gather*} e^{16 (9-4 x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(104976 - 186624*x + 124416*x^2 - 36864*x^3 + 4096*x^4)*(-186624 + 248832*x - 110592*x^2 + 16384*x
^3),x]

[Out]

E^(16*(9 - 4*x)^4)

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fricas [A]  time = 0.78, size = 21, normalized size = 1.05 \begin {gather*} e^{\left (4096 \, x^{4} - 36864 \, x^{3} + 124416 \, x^{2} - 186624 \, x + 104976\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16384*x^3-110592*x^2+248832*x-186624)*exp(4096*x^4-36864*x^3+124416*x^2-186624*x+104976),x, algorit
hm="fricas")

[Out]

e^(4096*x^4 - 36864*x^3 + 124416*x^2 - 186624*x + 104976)

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giac [A]  time = 0.18, size = 21, normalized size = 1.05 \begin {gather*} e^{\left (4096 \, x^{4} - 36864 \, x^{3} + 124416 \, x^{2} - 186624 \, x + 104976\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16384*x^3-110592*x^2+248832*x-186624)*exp(4096*x^4-36864*x^3+124416*x^2-186624*x+104976),x, algorit
hm="giac")

[Out]

e^(4096*x^4 - 36864*x^3 + 124416*x^2 - 186624*x + 104976)

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maple [A]  time = 2.87, size = 11, normalized size = 0.55

method	result	size
risch	\({\mathrm e}^{16 \left (4 x -9\right )^{4}}\)	\(11\)
gosper	\({\mathrm e}^{4096 x^{4}-36864 x^{3}+124416 x^{2}-186624 x +104976}\)	\(22\)
derivativedivides	\({\mathrm e}^{4096 x^{4}-36864 x^{3}+124416 x^{2}-186624 x +104976}\)	\(22\)
norman	\({\mathrm e}^{4096 x^{4}-36864 x^{3}+124416 x^{2}-186624 x +104976}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16384*x^3-110592*x^2+248832*x-186624)*exp(4096*x^4-36864*x^3+124416*x^2-186624*x+104976),x,method=_RETURN
VERBOSE)

[Out]

exp(16*(4*x-9)^4)

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maxima [A]  time = 0.36, size = 21, normalized size = 1.05 \begin {gather*} e^{\left (4096 \, x^{4} - 36864 \, x^{3} + 124416 \, x^{2} - 186624 \, x + 104976\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16384*x^3-110592*x^2+248832*x-186624)*exp(4096*x^4-36864*x^3+124416*x^2-186624*x+104976),x, algorit
hm="maxima")

[Out]

e^(4096*x^4 - 36864*x^3 + 124416*x^2 - 186624*x + 104976)

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mupad [B]  time = 0.07, size = 25, normalized size = 1.25 \begin {gather*} {\mathrm {e}}^{-186624\,x}\,{\mathrm {e}}^{104976}\,{\mathrm {e}}^{4096\,x^4}\,{\mathrm {e}}^{-36864\,x^3}\,{\mathrm {e}}^{124416\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(124416*x^2 - 186624*x - 36864*x^3 + 4096*x^4 + 104976)*(248832*x - 110592*x^2 + 16384*x^3 - 186624),x)

[Out]

exp(-186624*x)*exp(104976)*exp(4096*x^4)*exp(-36864*x^3)*exp(124416*x^2)

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sympy [A]  time = 0.12, size = 20, normalized size = 1.00 \begin {gather*} e^{4096 x^{4} - 36864 x^{3} + 124416 x^{2} - 186624 x + 104976} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16384*x**3-110592*x**2+248832*x-186624)*exp(4096*x**4-36864*x**3+124416*x**2-186624*x+104976),x)

[Out]

exp(4096*x**4 - 36864*x**3 + 124416*x**2 - 186624*x + 104976)

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