3.56.32 \(\int \frac {8 e+e^3 (2 x-2 e x)}{e^4} \, dx\)

Optimal. Leaf size=25 \[ -3+\frac {2 \left (e^e+4 x\right )}{e^3}+x \left (-x+\frac {x}{e}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 0.76, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {12} \begin {gather*} \frac {(1-e) x^2}{e}+\frac {8 x}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*E + E^3*(2*x - 2*E*x))/E^4,x]

[Out]

(8*x)/E^3 + ((1 - E)*x^2)/E

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (8 e+e^3 (2 x-2 e x)\right ) \, dx}{e^4}\\ &=\frac {8 x}{e^3}+\frac {(1-e) x^2}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.76 \begin {gather*} \frac {8 x}{e^3}-x^2+\frac {x^2}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*E + E^3*(2*x - 2*E*x))/E^4,x]

[Out]

(8*x)/E^3 - x^2 + x^2/E

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fricas [A]  time = 0.56, size = 21, normalized size = 0.84 \begin {gather*} -{\left (x^{2} e^{3} - x^{2} e^{2} - 8 \, x\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)+2*x)*exp(3)+8*exp(1))/exp(1)/exp(3),x, algorithm="fricas")

[Out]

-(x^2*e^3 - x^2*e^2 - 8*x)*e^(-3)

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giac [A]  time = 0.17, size = 25, normalized size = 1.00 \begin {gather*} -{\left ({\left (x^{2} e - x^{2}\right )} e^{3} - 8 \, x e\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)+2*x)*exp(3)+8*exp(1))/exp(1)/exp(3),x, algorithm="giac")

[Out]

-((x^2*e - x^2)*e^3 - 8*x*e)*e^(-4)

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maple [A]  time = 0.03, size = 22, normalized size = 0.88

method	result	size
norman	\(8 \,{\mathrm e}^{-3} x -\left ({\mathrm e}-1\right ) {\mathrm e}^{-1} x^{2}\)	\(22\)
risch	\({\mathrm e}^{-3} {\mathrm e}^{2} x^{2}-{\mathrm e}^{-3} x^{2} {\mathrm e}^{3}+8 \,{\mathrm e}^{-3} x\)	\(24\)
gosper	\(-x \left (x \,{\mathrm e} \,{\mathrm e}^{3}-x \,{\mathrm e}^{3}-8 \,{\mathrm e}\right ) {\mathrm e}^{-1} {\mathrm e}^{-3}\)	\(28\)
default	\({\mathrm e}^{-1} {\mathrm e}^{-3} \left ({\mathrm e}^{3} \left (-x^{2} {\mathrm e}+x^{2}\right )+8 x \,{\mathrm e}\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*exp(1)+2*x)*exp(3)+8*exp(1))/exp(1)/exp(3),x,method=_RETURNVERBOSE)

[Out]

8*x/exp(3)-(exp(1)-1)/exp(1)*x^2

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maxima [A]  time = 0.37, size = 25, normalized size = 1.00 \begin {gather*} -{\left ({\left (x^{2} e - x^{2}\right )} e^{3} - 8 \, x e\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)+2*x)*exp(3)+8*exp(1))/exp(1)/exp(3),x, algorithm="maxima")

[Out]

-((x^2*e - x^2)*e^3 - 8*x*e)*e^(-4)

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mupad [B]  time = 3.37, size = 14, normalized size = 0.56 \begin {gather*} \left ({\mathrm {e}}^{-1}-1\right )\,x^2+8\,{\mathrm {e}}^{-3}\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(8*exp(1) + exp(3)*(2*x - 2*x*exp(1))),x)

[Out]

8*x*exp(-3) + x^2*(exp(-1) - 1)

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sympy [A]  time = 0.05, size = 17, normalized size = 0.68 \begin {gather*} \frac {x^{2} \left (1 - e\right )}{e} + \frac {8 x}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)+2*x)*exp(3)+8*exp(1))/exp(1)/exp(3),x)

[Out]

x**2*(1 - E)*exp(-1) + 8*x*exp(-3)

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