3.1.62 \(\int \frac {1}{3} (-2+2 e^{5/3}+2 x) \, dx\)

Optimal. Leaf size=18 \[ \frac {1}{3} \left (1-e^{5/3}-x\right )^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {9} \begin {gather*} \frac {1}{3} \left (-x-e^{5/3}+1\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + 2*E^(5/3) + 2*x)/3,x]

[Out]

(1 - E^(5/3) - x)^2/3

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}{3} \left (1-e^{5/3}-x\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.22 \begin {gather*} \frac {2}{3} \left (-x+e^{5/3} x+\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 2*E^(5/3) + 2*x)/3,x]

[Out]

(2*(-x + E^(5/3)*x + x^2/2))/3

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fricas [A]  time = 0.65, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{3} \, x^{2} + \frac {2}{3} \, x e^{\frac {5}{3}} - \frac {2}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*exp(5/3)+2/3*x-2/3,x, algorithm="fricas")

[Out]

1/3*x^2 + 2/3*x*e^(5/3) - 2/3*x

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giac [A]  time = 0.24, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{3} \, x^{2} + \frac {2}{3} \, x e^{\frac {5}{3}} - \frac {2}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*exp(5/3)+2/3*x-2/3,x, algorithm="giac")

[Out]

1/3*x^2 + 2/3*x*e^(5/3) - 2/3*x

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maple [A]  time = 0.02, size = 11, normalized size = 0.61

method	result	size
gosper	\(\frac {x \left (x +2 \,{\mathrm e}^{\frac {5}{3}}-2\right )}{3}\)	\(11\)
default	\(\frac {2 x \,{\mathrm e}^{\frac {5}{3}}}{3}+\frac {x^{2}}{3}-\frac {2 x}{3}\)	\(15\)
norman	\(\left (\frac {2 \,{\mathrm e}^{\frac {5}{3}}}{3}-\frac {2}{3}\right ) x +\frac {x^{2}}{3}\)	\(15\)
risch	\(\frac {2 x \,{\mathrm e}^{\frac {5}{3}}}{3}+\frac {x^{2}}{3}-\frac {2 x}{3}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/3*exp(5/3)+2/3*x-2/3,x,method=_RETURNVERBOSE)

[Out]

1/3*x*(x+2*exp(5/3)-2)

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maxima [A]  time = 0.46, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{3} \, x^{2} + \frac {2}{3} \, x e^{\frac {5}{3}} - \frac {2}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*exp(5/3)+2/3*x-2/3,x, algorithm="maxima")

[Out]

1/3*x^2 + 2/3*x*e^(5/3) - 2/3*x

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mupad [B]  time = 0.04, size = 10, normalized size = 0.56 \begin {gather*} \frac {x\,\left (x+2\,{\mathrm {e}}^{5/3}-2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x)/3 + (2*exp(5/3))/3 - 2/3,x)

[Out]

(x*(x + 2*exp(5/3) - 2))/3

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sympy [A]  time = 0.05, size = 17, normalized size = 0.94 \begin {gather*} \frac {x^{2}}{3} + x \left (- \frac {2}{3} + \frac {2 e^{\frac {5}{3}}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*exp(5/3)+2/3*x-2/3,x)

[Out]

x**2/3 + x*(-2/3 + 2*exp(5/3)/3)

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