3.66.46 \(\int \frac {1}{3} (-e^2-4 x) \, dx\)

Optimal. Leaf size=16 \[ \frac {19}{3}-\frac {1}{3} x \left (e^2+2 x\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {9} \begin {gather*} -\frac {1}{24} \left (4 x+e^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^2 - 4*x)/3,x]

[Out]

-1/24*(E^2 + 4*x)^2

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {1}{24} \left (e^2+4 x\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {1}{3} \left (-e^2 x-2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^2 - 4*x)/3,x]

[Out]

(-(E^2*x) - 2*x^2)/3

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fricas [A]  time = 0.76, size = 11, normalized size = 0.69 \begin {gather*} -\frac {2}{3} \, x^{2} - \frac {1}{3} \, x e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3*exp(2)-4/3*x,x, algorithm="fricas")

[Out]

-2/3*x^2 - 1/3*x*e^2

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giac [A]  time = 0.22, size = 11, normalized size = 0.69 \begin {gather*} -\frac {2}{3} \, x^{2} - \frac {1}{3} \, x e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3*exp(2)-4/3*x,x, algorithm="giac")

[Out]

-2/3*x^2 - 1/3*x*e^2

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maple [A]  time = 0.05, size = 10, normalized size = 0.62

method	result	size
gosper	\(-\frac {\left ({\mathrm e}^{2}+2 x \right ) x}{3}\)	\(10\)
default	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)
norman	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)
risch	\(-\frac {{\mathrm e}^{2} x}{3}-\frac {2 x^{2}}{3}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/3*exp(2)-4/3*x,x,method=_RETURNVERBOSE)

[Out]

-1/3*(exp(2)+2*x)*x

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maxima [A]  time = 0.42, size = 11, normalized size = 0.69 \begin {gather*} -\frac {2}{3} \, x^{2} - \frac {1}{3} \, x e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3*exp(2)-4/3*x,x, algorithm="maxima")

[Out]

-2/3*x^2 - 1/3*x*e^2

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mupad [B]  time = 0.03, size = 9, normalized size = 0.56 \begin {gather*} -\frac {x\,\left (2\,x+{\mathrm {e}}^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- (4*x)/3 - exp(2)/3,x)

[Out]

-(x*(2*x + exp(2)))/3

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sympy [A]  time = 0.05, size = 14, normalized size = 0.88 \begin {gather*} - \frac {2 x^{2}}{3} - \frac {x e^{2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3*exp(2)-4/3*x,x)

[Out]

-2*x**2/3 - x*exp(2)/3

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