3.3.38 \(\int \frac {1568 x-5040 x^2+4048 x^3-600 x^4+24 x^5+e^4 (2+2 x)+e^2 (-56+8 x+168 x^2-16 x^3)}{e^4} \, dx\)

Optimal. Leaf size=22 \[ \left (-1-x+\frac {2 (-14+x) \left (-x+x^2\right )}{e^2}\right )^2 \]

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Rubi [B]  time = 0.03, antiderivative size = 76, normalized size of antiderivative = 3.45, number of steps used = 3, number of rules used = 1, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {12} \begin {gather*} \frac {4 x^6}{e^4}-\frac {120 x^5}{e^4}-\frac {4 x^4}{e^2}+\frac {1012 x^4}{e^4}+\frac {56 x^3}{e^2}-\frac {1680 x^3}{e^4}+\frac {4 x^2}{e^2}+\frac {784 x^2}{e^4}-\frac {56 x}{e^2}+(x+1)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1568*x - 5040*x^2 + 4048*x^3 - 600*x^4 + 24*x^5 + E^4*(2 + 2*x) + E^2*(-56 + 8*x + 168*x^2 - 16*x^3))/E^4
,x]

[Out]

(-56*x)/E^2 + (784*x^2)/E^4 + (4*x^2)/E^2 - (1680*x^3)/E^4 + (56*x^3)/E^2 + (1012*x^4)/E^4 - (4*x^4)/E^2 - (12
0*x^5)/E^4 + (4*x^6)/E^4 + (1 + x)^2

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 (1568 x-5040 x^2+4048 x^3-600 x^4+24 x^5+e^4 (2+2 x)+e^2 \left (-56+8 x+168 x^2-16 x^3\right )\right ) \, dx}{e^4}\\ &=\frac {784 x^2}{e^4}-\frac {1680 x^3}{e^4}+\frac {1012 x^4}{e^4}-\frac {120 x^5}{e^4}+\frac {4 x^6}{e^4}+(1+x)^2+\frac {\int \left (-56+8 x+168 x^2-16 x^3\right ) \, dx}{e^2}\\ &=-\frac {56 x}{e^2}+\frac {784 x^2}{e^4}+\frac {4 x^2}{e^2}-\frac {1680 x^3}{e^4}+\frac {56 x^3}{e^2}+\frac {1012 x^4}{e^4}-\frac {4 x^4}{e^2}-\frac {120 x^5}{e^4}+\frac {4 x^6}{e^4}+(1+x)^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 81, normalized size = 3.68 \begin {gather*} 2 \left (x-\frac {28 x}{e^2}+\frac {x^2}{2}+\frac {392 x^2}{e^4}+\frac {2 x^2}{e^2}-\frac {840 x^3}{e^4}+\frac {28 x^3}{e^2}+\frac {506 x^4}{e^4}-\frac {2 x^4}{e^2}-\frac {60 x^5}{e^4}+\frac {2 x^6}{e^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1568*x - 5040*x^2 + 4048*x^3 - 600*x^4 + 24*x^5 + E^4*(2 + 2*x) + E^2*(-56 + 8*x + 168*x^2 - 16*x^3
))/E^4,x]

[Out]

2*(x - (28*x)/E^2 + x^2/2 + (392*x^2)/E^4 + (2*x^2)/E^2 - (840*x^3)/E^4 + (28*x^3)/E^2 + (506*x^4)/E^4 - (2*x^
4)/E^2 - (60*x^5)/E^4 + (2*x^6)/E^4)

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fricas [B]  time = 0.63, size = 60, normalized size = 2.73 \begin {gather*} {\left (4 \, x^{6} - 120 \, x^{5} + 1012 \, x^{4} - 1680 \, x^{3} + 784 \, x^{2} + {\left (x^{2} + 2 \, x\right )} e^{4} - 4 \, {\left (x^{4} - 14 \, x^{3} - x^{2} + 14 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*exp(2)^2+(-16*x^3+168*x^2+8*x-56)*exp(2)+24*x^5-600*x^4+4048*x^3-5040*x^2+1568*x)/exp(2)^2,
x, algorithm="fricas")

[Out]

(4*x^6 - 120*x^5 + 1012*x^4 - 1680*x^3 + 784*x^2 + (x^2 + 2*x)*e^4 - 4*(x^4 - 14*x^3 - x^2 + 14*x)*e^2)*e^(-4)

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giac [B]  time = 0.36, size = 60, normalized size = 2.73 \begin {gather*} {\left (4 \, x^{6} - 120 \, x^{5} + 1012 \, x^{4} - 1680 \, x^{3} + 784 \, x^{2} + {\left (x^{2} + 2 \, x\right )} e^{4} - 4 \, {\left (x^{4} - 14 \, x^{3} - x^{2} + 14 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*exp(2)^2+(-16*x^3+168*x^2+8*x-56)*exp(2)+24*x^5-600*x^4+4048*x^3-5040*x^2+1568*x)/exp(2)^2,
x, algorithm="giac")

[Out]

(4*x^6 - 120*x^5 + 1012*x^4 - 1680*x^3 + 784*x^2 + (x^2 + 2*x)*e^4 - 4*(x^4 - 14*x^3 - x^2 + 14*x)*e^2)*e^(-4)

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maple [B]  time = 0.10, size = 66, normalized size = 3.00

method	result	size
gosper	\(x \left (4 x^{5}-4 x^{3} {\mathrm e}^{2}-120 x^{4}+x \,{\mathrm e}^{4}+56 x^{2} {\mathrm e}^{2}+1012 x^{3}+2 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2} x -1680 x^{2}-56 \,{\mathrm e}^{2}+784 x \right ) {\mathrm e}^{-4}\)	\(66\)
default	\({\mathrm e}^{-4} \left (\left (x^{2}+2 x \right ) {\mathrm e}^{4}+{\mathrm e}^{2} \left (-4 x^{4}+56 x^{3}+4 x^{2}-56 x \right )+4 x^{6}-120 x^{5}+1012 x^{4}-1680 x^{3}+784 x^{2}\right )\)	\(66\)
risch	\(4 \,{\mathrm e}^{-4} x^{6}-4 \,{\mathrm e}^{-2} x^{4}-120 \,{\mathrm e}^{-4} x^{5}+x^{2}+56 \,{\mathrm e}^{-2} x^{3}+1012 \,{\mathrm e}^{-4} x^{4}+2 x +4 x^{2} {\mathrm e}^{-2}-1680 \,{\mathrm e}^{-4} x^{3}-56 x \,{\mathrm e}^{-2}+784 \,{\mathrm e}^{-4} x^{2}\)	\(69\)
norman	\(\left (\left (2 \,{\mathrm e}^{2}-56\right ) x +\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+784\right ) {\mathrm e}^{-2} x^{2}-120 \,{\mathrm e}^{-2} x^{5}+4 \,{\mathrm e}^{-2} x^{6}-4 \left ({\mathrm e}^{2}-253\right ) {\mathrm e}^{-2} x^{4}+56 \left ({\mathrm e}^{2}-30\right ) {\mathrm e}^{-2} x^{3}\right ) {\mathrm e}^{-2}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x+2)*exp(2)^2+(-16*x^3+168*x^2+8*x-56)*exp(2)+24*x^5-600*x^4+4048*x^3-5040*x^2+1568*x)/exp(2)^2,x,meth
od=_RETURNVERBOSE)

[Out]

x*(4*x^5-4*x^3*exp(2)-120*x^4+x*exp(2)^2+56*x^2*exp(2)+1012*x^3+2*exp(2)^2+4*exp(2)*x-1680*x^2-56*exp(2)+784*x
)/exp(2)^2

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maxima [B]  time = 0.40, size = 60, normalized size = 2.73 \begin {gather*} {\left (4 \, x^{6} - 120 \, x^{5} + 1012 \, x^{4} - 1680 \, x^{3} + 784 \, x^{2} + {\left (x^{2} + 2 \, x\right )} e^{4} - 4 \, {\left (x^{4} - 14 \, x^{3} - x^{2} + 14 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*exp(2)^2+(-16*x^3+168*x^2+8*x-56)*exp(2)+24*x^5-600*x^4+4048*x^3-5040*x^2+1568*x)/exp(2)^2,
x, algorithm="maxima")

[Out]

(4*x^6 - 120*x^5 + 1012*x^4 - 1680*x^3 + 784*x^2 + (x^2 + 2*x)*e^4 - 4*(x^4 - 14*x^3 - x^2 + 14*x)*e^2)*e^(-4)

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mupad [B]  time = 0.33, size = 67, normalized size = 3.05 \begin {gather*} 4\,{\mathrm {e}}^{-4}\,x^6-120\,{\mathrm {e}}^{-4}\,x^5-\frac {{\mathrm {e}}^{-4}\,\left (16\,{\mathrm {e}}^2-4048\right )\,x^4}{4}+\frac {{\mathrm {e}}^{-4}\,\left (168\,{\mathrm {e}}^2-5040\right )\,x^3}{3}+\frac {{\mathrm {e}}^{-4}\,\left (8\,{\mathrm {e}}^2+2\,{\mathrm {e}}^4+1568\right )\,x^2}{2}+2\,{\mathrm {e}}^{-2}\,\left ({\mathrm {e}}^2-28\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(1568*x + exp(2)*(8*x + 168*x^2 - 16*x^3 - 56) - 5040*x^2 + 4048*x^3 - 600*x^4 + 24*x^5 + exp(4)*(
2*x + 2)),x)

[Out]

4*x^6*exp(-4) - 120*x^5*exp(-4) + 2*x*exp(-2)*(exp(2) - 28) + (x^2*exp(-4)*(8*exp(2) + 2*exp(4) + 1568))/2 - (
x^4*exp(-4)*(16*exp(2) - 4048))/4 + (x^3*exp(-4)*(168*exp(2) - 5040))/3

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sympy [B]  time = 0.07, size = 71, normalized size = 3.23 \begin {gather*} \frac {4 x^{6}}{e^{4}} - \frac {120 x^{5}}{e^{4}} + \frac {x^{4} \left (1012 - 4 e^{2}\right )}{e^{4}} + \frac {x^{3} \left (-1680 + 56 e^{2}\right )}{e^{4}} + \frac {x^{2} \left (4 e^{2} + e^{4} + 784\right )}{e^{4}} + \frac {x \left (-56 + 2 e^{2}\right )}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*exp(2)**2+(-16*x**3+168*x**2+8*x-56)*exp(2)+24*x**5-600*x**4+4048*x**3-5040*x**2+1568*x)/ex
p(2)**2,x)

[Out]

4*x**6*exp(-4) - 120*x**5*exp(-4) + x**4*(1012 - 4*exp(2))*exp(-4) + x**3*(-1680 + 56*exp(2))*exp(-4) + x**2*(
4*exp(2) + exp(4) + 784)*exp(-4) + x*(-56 + 2*exp(2))*exp(-2)

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