3.41.47 \(\int (8+6 e^{6 x}-4 x) \, dx\)

Optimal. Leaf size=13 \[ e^{6 x}-2 (-2+x)^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2194} \begin {gather*} -2 x^2+8 x+e^{6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8 + 6*E^(6*x) - 4*x,x]

[Out]

E^(6*x) + 8*x - 2*x^2

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} &=8 x-2 x^2+6 \int e^{6 x} \, dx\\ &=e^{6 x}+8 x-2 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.08 \begin {gather*} e^{6 x}+8 x-2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8 + 6*E^(6*x) - 4*x,x]

[Out]

E^(6*x) + 8*x - 2*x^2

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fricas [A]  time = 1.24, size = 13, normalized size = 1.00 \begin {gather*} -2 \, x^{2} + 8 \, x + e^{\left (6 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(6*x)-4*x+8,x, algorithm="fricas")

[Out]

-2*x^2 + 8*x + e^(6*x)

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giac [A]  time = 0.19, size = 13, normalized size = 1.00 \begin {gather*} -2 \, x^{2} + 8 \, x + e^{\left (6 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(6*x)-4*x+8,x, algorithm="giac")

[Out]

-2*x^2 + 8*x + e^(6*x)

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maple [A]  time = 0.02, size = 14, normalized size = 1.08

method	result	size
derivativedivides	\(8 x -2 x^{2}+{\mathrm e}^{6 x}\)	\(14\)
default	\(8 x -2 x^{2}+{\mathrm e}^{6 x}\)	\(14\)
norman	\(8 x -2 x^{2}+{\mathrm e}^{6 x}\)	\(14\)
risch	\(8 x -2 x^{2}+{\mathrm e}^{6 x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*exp(6*x)-4*x+8,x,method=_RETURNVERBOSE)

[Out]

8*x-2*x^2+exp(6*x)

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maxima [A]  time = 0.36, size = 13, normalized size = 1.00 \begin {gather*} -2 \, x^{2} + 8 \, x + e^{\left (6 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(6*x)-4*x+8,x, algorithm="maxima")

[Out]

-2*x^2 + 8*x + e^(6*x)

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mupad [B]  time = 2.99, size = 13, normalized size = 1.00 \begin {gather*} 8\,x+{\mathrm {e}}^{6\,x}-2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*exp(6*x) - 4*x + 8,x)

[Out]

8*x + exp(6*x) - 2*x^2

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sympy [A]  time = 0.07, size = 12, normalized size = 0.92 \begin {gather*} - 2 x^{2} + 8 x + e^{6 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*exp(6*x)-4*x+8,x)

[Out]

-2*x**2 + 8*x + exp(6*x)

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