3.62.37 \(\int (1+e^{8+x-2 x^2} (-1+4 x)) \, dx\)

Optimal. Leaf size=15 \[ -4-e^{8+x-2 x^2}+x \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2236} \begin {gather*} x-e^{-2 x^2+x+8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^(8 + x - 2*x^2)*(-1 + 4*x),x]

[Out]

-E^(8 + x - 2*x^2) + x

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+\int e^{8+x-2 x^2} (-1+4 x) \, dx\\ &=-e^{8+x-2 x^2}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 14, normalized size = 0.93 \begin {gather*} -e^{8+x-2 x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(8 + x - 2*x^2)*(-1 + 4*x),x]

[Out]

-E^(8 + x - 2*x^2) + x

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fricas [A]  time = 0.79, size = 13, normalized size = 0.87 \begin {gather*} x - e^{\left (-2 \, x^{2} + x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-1)*exp(-2*x^2+x+8)+1,x, algorithm="fricas")

[Out]

x - e^(-2*x^2 + x + 8)

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giac [A]  time = 0.13, size = 13, normalized size = 0.87 \begin {gather*} x - e^{\left (-2 \, x^{2} + x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-1)*exp(-2*x^2+x+8)+1,x, algorithm="giac")

[Out]

x - e^(-2*x^2 + x + 8)

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maple [A]  time = 0.04, size = 14, normalized size = 0.93

method	result	size
default	\(x -{\mathrm e}^{-2 x^{2}+x +8}\)	\(14\)
norman	\(x -{\mathrm e}^{-2 x^{2}+x +8}\)	\(14\)
risch	\(x -{\mathrm e}^{-2 x^{2}+x +8}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x-1)*exp(-2*x^2+x+8)+1,x,method=_RETURNVERBOSE)

[Out]

x-exp(-2*x^2+x+8)

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maxima [A]  time = 0.35, size = 13, normalized size = 0.87 \begin {gather*} x - e^{\left (-2 \, x^{2} + x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-1)*exp(-2*x^2+x+8)+1,x, algorithm="maxima")

[Out]

x - e^(-2*x^2 + x + 8)

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mupad [B]  time = 4.38, size = 14, normalized size = 0.93 \begin {gather*} x-{\mathrm {e}}^8\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x - 2*x^2 + 8)*(4*x - 1) + 1,x)

[Out]

x - exp(8)*exp(-2*x^2)*exp(x)

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sympy [A]  time = 0.08, size = 10, normalized size = 0.67 \begin {gather*} x - e^{- 2 x^{2} + x + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-1)*exp(-2*x**2+x+8)+1,x)

[Out]

x - exp(-2*x**2 + x + 8)

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