3.58.22 \(\int \frac {1}{2} (2+e^{\frac {1}{2} (4959+543 x-17 x^2-2 x^3)} (543-34 x-6 x^2)) \, dx\)

Optimal. Leaf size=21 \[ 5+e^{x+\left (\frac {19}{2}+x\right ) \left (261+x-x^2\right )}+x \]

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Rubi [A]  time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{2} \left (-2 x^3-17 x^2+543 x+4959\right )}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^((4959 + 543*x - 17*x^2 - 2*x^3)/2)*(543 - 34*x - 6*x^2))/2,x]

[Out]

E^((4959 + 543*x - 17*x^2 - 2*x^3)/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}{2} \int \left (2+e^{\frac {1}{2} \left (4959+543 x-17 x^2-2 x^3\right )} \left (543-34 x-6 x^2\right )\right ) \, dx\\ &=x+\frac {1}{2} \int e^{\frac {1}{2} \left (4959+543 x-17 x^2-2 x^3\right )} \left (543-34 x-6 x^2\right ) \, dx\\ &=e^{\frac {1}{2} \left (4959+543 x-17 x^2-2 x^3\right )}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 25, normalized size = 1.19 \begin {gather*} e^{\frac {4959}{2}+\frac {543 x}{2}-\frac {17 x^2}{2}-x^3}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^((4959 + 543*x - 17*x^2 - 2*x^3)/2)*(543 - 34*x - 6*x^2))/2,x]

[Out]

E^(4959/2 + (543*x)/2 - (17*x^2)/2 - x^3) + x

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fricas [A]  time = 0.64, size = 18, normalized size = 0.86 \begin {gather*} x + e^{\left (-x^{3} - \frac {17}{2} \, x^{2} + \frac {543}{2} \, x + \frac {4959}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-6*x^2-34*x+543)*exp(-x^3-17/2*x^2+543/2*x+4959/2)+1,x, algorithm="fricas")

[Out]

x + e^(-x^3 - 17/2*x^2 + 543/2*x + 4959/2)

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giac [A]  time = 0.15, size = 18, normalized size = 0.86 \begin {gather*} x + e^{\left (-x^{3} - \frac {17}{2} \, x^{2} + \frac {543}{2} \, x + \frac {4959}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-6*x^2-34*x+543)*exp(-x^3-17/2*x^2+543/2*x+4959/2)+1,x, algorithm="giac")

[Out]

x + e^(-x^3 - 17/2*x^2 + 543/2*x + 4959/2)

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maple [A]  time = 0.06, size = 19, normalized size = 0.90

method	result	size
default	\(x +{\mathrm e}^{-x^{3}-\frac {17}{2} x^{2}+\frac {543}{2} x +\frac {4959}{2}}\)	\(19\)
norman	\(x +{\mathrm e}^{-x^{3}-\frac {17}{2} x^{2}+\frac {543}{2} x +\frac {4959}{2}}\)	\(19\)
risch	\(x +{\mathrm e}^{-x^{3}-\frac {17}{2} x^{2}+\frac {543}{2} x +\frac {4959}{2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-6*x^2-34*x+543)*exp(-x^3-17/2*x^2+543/2*x+4959/2)+1,x,method=_RETURNVERBOSE)

[Out]

x+exp(-x^3-17/2*x^2+543/2*x+4959/2)

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maxima [A]  time = 0.35, size = 18, normalized size = 0.86 \begin {gather*} x + e^{\left (-x^{3} - \frac {17}{2} \, x^{2} + \frac {543}{2} \, x + \frac {4959}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-6*x^2-34*x+543)*exp(-x^3-17/2*x^2+543/2*x+4959/2)+1,x, algorithm="maxima")

[Out]

x + e^(-x^3 - 17/2*x^2 + 543/2*x + 4959/2)

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mupad [B]  time = 0.09, size = 18, normalized size = 0.86 \begin {gather*} x+{\mathrm {e}}^{-x^3-\frac {17\,x^2}{2}+\frac {543\,x}{2}+\frac {4959}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - (exp((543*x)/2 - (17*x^2)/2 - x^3 + 4959/2)*(34*x + 6*x^2 - 543))/2,x)

[Out]

x + exp((543*x)/2 - (17*x^2)/2 - x^3 + 4959/2)

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sympy [A]  time = 0.11, size = 20, normalized size = 0.95 \begin {gather*} x + e^{- x^{3} - \frac {17 x^{2}}{2} + \frac {543 x}{2} + \frac {4959}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-6*x**2-34*x+543)*exp(-x**3-17/2*x**2+543/2*x+4959/2)+1,x)

[Out]

x + exp(-x**3 - 17*x**2/2 + 543*x/2 + 4959/2)

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