3.81.79 \(\int \frac {1}{80} (5 x^4-12 x^5+7 x^6+e^5 (-16 x+48 x^2-32 x^3)) \, dx\)

Optimal. Leaf size=32 \[ \frac {1}{5} \left (3+\frac {1}{4} \left (x-x^2\right )^2 \left (-2 e^5+\frac {x^3}{4}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.62, number of steps used = 3, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {12} \begin {gather*} \frac {x^7}{80}-\frac {x^6}{40}+\frac {x^5}{80}-\frac {e^5 x^4}{10}+\frac {e^5 x^3}{5}-\frac {e^5 x^2}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*x^4 - 12*x^5 + 7*x^6 + E^5*(-16*x + 48*x^2 - 32*x^3))/80,x]

[Out]

-1/10*(E^5*x^2) + (E^5*x^3)/5 - (E^5*x^4)/10 + x^5/80 - x^6/40 + x^7/80

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 {1}{80} \int \left (5 x^4-12 x^5+7 x^6+e^5 \left (-16 x+48 x^2-32 x^3\right )\right ) \, dx\\ &=\frac {x^5}{80}-\frac {x^6}{40}+\frac {x^7}{80}+\frac {1}{80} e^5 \int \left (-16 x+48 x^2-32 x^3\right ) \, dx\\ &=-\frac {1}{10} e^5 x^2+\frac {e^5 x^3}{5}-\frac {e^5 x^4}{10}+\frac {x^5}{80}-\frac {x^6}{40}+\frac {x^7}{80}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 1.25 \begin {gather*} \frac {1}{80} \left (-8 e^5 x^2+16 e^5 x^3-8 e^5 x^4+x^5-2 x^6+x^7\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*x^4 - 12*x^5 + 7*x^6 + E^5*(-16*x + 48*x^2 - 32*x^3))/80,x]

[Out]

(-8*E^5*x^2 + 16*E^5*x^3 - 8*E^5*x^4 + x^5 - 2*x^6 + x^7)/80

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fricas [A]  time = 0.54, size = 32, normalized size = 1.00 \begin {gather*} \frac {1}{80} \, x^{7} - \frac {1}{40} \, x^{6} + \frac {1}{80} \, x^{5} - \frac {1}{10} \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(-32*x^3+48*x^2-16*x)*exp(5)+7/80*x^6-3/20*x^5+1/16*x^4,x, algorithm="fricas")

[Out]

1/80*x^7 - 1/40*x^6 + 1/80*x^5 - 1/10*(x^4 - 2*x^3 + x^2)*e^5

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giac [A]  time = 0.18, size = 32, normalized size = 1.00 \begin {gather*} \frac {1}{80} \, x^{7} - \frac {1}{40} \, x^{6} + \frac {1}{80} \, x^{5} - \frac {1}{10} \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(-32*x^3+48*x^2-16*x)*exp(5)+7/80*x^6-3/20*x^5+1/16*x^4,x, algorithm="giac")

[Out]

1/80*x^7 - 1/40*x^6 + 1/80*x^5 - 1/10*(x^4 - 2*x^3 + x^2)*e^5

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maple [A]  time = 0.04, size = 27, normalized size = 0.84

method	result	size
gosper	\(-\frac {x^{2} \left (-x^{4}+x^{3}+8 x \,{\mathrm e}^{5}-8 \,{\mathrm e}^{5}\right ) \left (x -1\right )}{80}\)	\(27\)
default	\(\frac {{\mathrm e}^{5} \left (-8 x^{4}+16 x^{3}-8 x^{2}\right )}{80}+\frac {x^{7}}{80}-\frac {x^{6}}{40}+\frac {x^{5}}{80}\)	\(37\)
norman	\(\frac {x^{5}}{80}-\frac {x^{6}}{40}+\frac {x^{7}}{80}-\frac {x^{2} {\mathrm e}^{5}}{10}+\frac {x^{3} {\mathrm e}^{5}}{5}-\frac {x^{4} {\mathrm e}^{5}}{10}\)	\(38\)
risch	\(\frac {x^{5}}{80}-\frac {x^{6}}{40}+\frac {x^{7}}{80}-\frac {x^{2} {\mathrm e}^{5}}{10}+\frac {x^{3} {\mathrm e}^{5}}{5}-\frac {x^{4} {\mathrm e}^{5}}{10}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/80*(-32*x^3+48*x^2-16*x)*exp(5)+7/80*x^6-3/20*x^5+1/16*x^4,x,method=_RETURNVERBOSE)

[Out]

-1/80*x^2*(-x^4+x^3+8*x*exp(5)-8*exp(5))*(x-1)

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maxima [A]  time = 0.36, size = 32, normalized size = 1.00 \begin {gather*} \frac {1}{80} \, x^{7} - \frac {1}{40} \, x^{6} + \frac {1}{80} \, x^{5} - \frac {1}{10} \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(-32*x^3+48*x^2-16*x)*exp(5)+7/80*x^6-3/20*x^5+1/16*x^4,x, algorithm="maxima")

[Out]

1/80*x^7 - 1/40*x^6 + 1/80*x^5 - 1/10*(x^4 - 2*x^3 + x^2)*e^5

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mupad [B]  time = 0.05, size = 37, normalized size = 1.16 \begin {gather*} \frac {x^7}{80}-\frac {x^6}{40}+\frac {x^5}{80}-\frac {{\mathrm {e}}^5\,x^4}{10}+\frac {{\mathrm {e}}^5\,x^3}{5}-\frac {{\mathrm {e}}^5\,x^2}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/16 - (exp(5)*(16*x - 48*x^2 + 32*x^3))/80 - (3*x^5)/20 + (7*x^6)/80,x)

[Out]

(x^3*exp(5))/5 - (x^2*exp(5))/10 - (x^4*exp(5))/10 + x^5/80 - x^6/40 + x^7/80

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sympy [A]  time = 0.06, size = 39, normalized size = 1.22 \begin {gather*} \frac {x^{7}}{80} - \frac {x^{6}}{40} + \frac {x^{5}}{80} - \frac {x^{4} e^{5}}{10} + \frac {x^{3} e^{5}}{5} - \frac {x^{2} e^{5}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(-32*x**3+48*x**2-16*x)*exp(5)+7/80*x**6-3/20*x**5+1/16*x**4,x)

[Out]

x**7/80 - x**6/40 + x**5/80 - x**4*exp(5)/10 + x**3*exp(5)/5 - x**2*exp(5)/10

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