3.46.74 \(\int e^x (-1293+864 e-216 e^2+24 e^3-e^4-x) \, dx\)

Optimal. Leaf size=16 \[ e^x \left (4-(-6+e)^4-x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.75, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2176, 2194} \begin {gather*} e^x-e^x \left (x+e^4-24 e^3+216 e^2-864 e+1293\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*(-1293 + 864*E - 216*E^2 + 24*E^3 - E^4 - x),x]

[Out]

E^x - E^x*(1293 - 864*E + 216*E^2 - 24*E^3 + E^4 + x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-e^x \left (1293-864 e+216 e^2-24 e^3+e^4+x\right )+\int e^x \, dx\\ &=e^x-e^x \left (1293-864 e+216 e^2-24 e^3+e^4+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 24, normalized size = 1.50 \begin {gather*} -e^x \left (1292-864 e+216 e^2-24 e^3+e^4+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*(-1293 + 864*E - 216*E^2 + 24*E^3 - E^4 - x),x]

[Out]

-(E^x*(1292 - 864*E + 216*E^2 - 24*E^3 + E^4 + x))

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fricas [A]  time = 0.64, size = 21, normalized size = 1.31 \begin {gather*} -{\left (x + e^{4} - 24 \, e^{3} + 216 \, e^{2} - 864 \, e + 1292\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1)^4+24*exp(1)^3-216*exp(1)^2+864*exp(1)-x-1293)*exp(x),x, algorithm="fricas")

[Out]

-(x + e^4 - 24*e^3 + 216*e^2 - 864*e + 1292)*e^x

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giac [B]  time = 0.12, size = 32, normalized size = 2.00 \begin {gather*} -{\left (x + 1292\right )} e^{x} - e^{\left (x + 4\right )} + 24 \, e^{\left (x + 3\right )} - 216 \, e^{\left (x + 2\right )} + 864 \, e^{\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1)^4+24*exp(1)^3-216*exp(1)^2+864*exp(1)-x-1293)*exp(x),x, algorithm="giac")

[Out]

-(x + 1292)*e^x - e^(x + 4) + 24*e^(x + 3) - 216*e^(x + 2) + 864*e^(x + 1)

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maple [A]  time = 0.03, size = 25, normalized size = 1.56

method	result	size
risch	\(\left (-{\mathrm e}^{4}+24 \,{\mathrm e}^{3}-216 \,{\mathrm e}^{2}+864 \,{\mathrm e}-x -1292\right ) {\mathrm e}^{x}\)	\(25\)
gosper	\(-{\mathrm e}^{x} \left (x +{\mathrm e}^{4}-24 \,{\mathrm e}^{3}+216 \,{\mathrm e}^{2}-864 \,{\mathrm e}+1292\right )\)	\(28\)
norman	\(\left (-{\mathrm e}^{4}+24 \,{\mathrm e}^{3}-216 \,{\mathrm e}^{2}+864 \,{\mathrm e}-1292\right ) {\mathrm e}^{x}-{\mathrm e}^{x} x\)	\(34\)
default	\(864 \,{\mathrm e} \,{\mathrm e}^{x}-216 \,{\mathrm e}^{2} {\mathrm e}^{x}-{\mathrm e}^{x} x -1292 \,{\mathrm e}^{x}+24 \,{\mathrm e}^{x} {\mathrm e}^{3}-{\mathrm e}^{4} {\mathrm e}^{x}\)	\(41\)
meijerg	\({\mathrm e}^{4} \left (1-{\mathrm e}^{x}\right )-24 \,{\mathrm e}^{3} \left (1-{\mathrm e}^{x}\right )+216 \,{\mathrm e}^{2} \left (1-{\mathrm e}^{x}\right )-864 \left (1-{\mathrm e}^{x}\right ) {\mathrm e}+1292+\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}-1293 \,{\mathrm e}^{x}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(1)^4+24*exp(1)^3-216*exp(1)^2+864*exp(1)-x-1293)*exp(x),x,method=_RETURNVERBOSE)

[Out]

(-exp(4)+24*exp(3)-216*exp(2)+864*exp(1)-x-1292)*exp(x)

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maxima [B]  time = 0.36, size = 36, normalized size = 2.25 \begin {gather*} -{\left (x - 1\right )} e^{x} - e^{\left (x + 4\right )} + 24 \, e^{\left (x + 3\right )} - 216 \, e^{\left (x + 2\right )} + 864 \, e^{\left (x + 1\right )} - 1293 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1)^4+24*exp(1)^3-216*exp(1)^2+864*exp(1)-x-1293)*exp(x),x, algorithm="maxima")

[Out]

-(x - 1)*e^x - e^(x + 4) + 24*e^(x + 3) - 216*e^(x + 2) + 864*e^(x + 1) - 1293*e^x

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mupad [B]  time = 3.07, size = 21, normalized size = 1.31 \begin {gather*} -{\mathrm {e}}^x\,\left (x-864\,\mathrm {e}+216\,{\mathrm {e}}^2-24\,{\mathrm {e}}^3+{\mathrm {e}}^4+1292\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x)*(x - 864*exp(1) + 216*exp(2) - 24*exp(3) + exp(4) + 1293),x)

[Out]

-exp(x)*(x - 864*exp(1) + 216*exp(2) - 24*exp(3) + exp(4) + 1292)

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sympy [A]  time = 0.10, size = 24, normalized size = 1.50 \begin {gather*} \left (- x - 216 e^{2} - 1292 - e^{4} + 24 e^{3} + 864 e\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1)**4+24*exp(1)**3-216*exp(1)**2+864*exp(1)-x-1293)*exp(x),x)

[Out]

(-x - 216*exp(2) - 1292 - exp(4) + 24*exp(3) + 864*E)*exp(x)

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