3.77.94 \(\int \frac {243 x^2+108 x^3-270 x^4-252 x^5-77 x^6-8 x^7}{e^{64/3}} \, dx\)

Optimal. Leaf size=19 \[ \frac {(1-x) x^3 (3+x)^4}{e^{64/3}} \]

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Rubi [B]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 3.21, number of steps used = 2, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {12} \begin {gather*} -\frac {x^8}{e^{64/3}}-\frac {11 x^7}{e^{64/3}}-\frac {42 x^6}{e^{64/3}}-\frac {54 x^5}{e^{64/3}}+\frac {27 x^4}{e^{64/3}}+\frac {81 x^3}{e^{64/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(243*x^2 + 108*x^3 - 270*x^4 - 252*x^5 - 77*x^6 - 8*x^7)/E^(64/3),x]

[Out]

(81*x^3)/E^(64/3) + (27*x^4)/E^(64/3) - (54*x^5)/E^(64/3) - (42*x^6)/E^(64/3) - (11*x^7)/E^(64/3) - x^8/E^(64/
3)

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 (243 x^2+108 x^3-270 x^4-252 x^5-77 x^6-8 x^7\right ) \, dx}{e^{64/3}}\\ &=\frac {81 x^3}{e^{64/3}}+\frac {27 x^4}{e^{64/3}}-\frac {54 x^5}{e^{64/3}}-\frac {42 x^6}{e^{64/3}}-\frac {11 x^7}{e^{64/3}}-\frac {x^8}{e^{64/3}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.89 \begin {gather*} -\frac {-81 x^3-27 x^4+54 x^5+42 x^6+11 x^7+x^8}{e^{64/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(243*x^2 + 108*x^3 - 270*x^4 - 252*x^5 - 77*x^6 - 8*x^7)/E^(64/3),x]

[Out]

-((-81*x^3 - 27*x^4 + 54*x^5 + 42*x^6 + 11*x^7 + x^8)/E^(64/3))

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fricas [B]  time = 0.60, size = 33, normalized size = 1.74 \begin {gather*} -{\left (x^{8} + 11 \, x^{7} + 42 \, x^{6} + 54 \, x^{5} - 27 \, x^{4} - 81 \, x^{3}\right )} e^{\left (-\frac {64}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^7-77*x^6-252*x^5-270*x^4+108*x^3+243*x^2)/exp(4/3)^4/exp(4)^4,x, algorithm="fricas")

[Out]

-(x^8 + 11*x^7 + 42*x^6 + 54*x^5 - 27*x^4 - 81*x^3)*e^(-64/3)

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giac [B]  time = 0.21, size = 33, normalized size = 1.74 \begin {gather*} -{\left (x^{8} + 11 \, x^{7} + 42 \, x^{6} + 54 \, x^{5} - 27 \, x^{4} - 81 \, x^{3}\right )} e^{\left (-\frac {64}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^7-77*x^6-252*x^5-270*x^4+108*x^3+243*x^2)/exp(4/3)^4/exp(4)^4,x, algorithm="giac")

[Out]

-(x^8 + 11*x^7 + 42*x^6 + 54*x^5 - 27*x^4 - 81*x^3)*e^(-64/3)

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maple [A]  time = 0.04, size = 37, normalized size = 1.95

method	result	size
gosper	\(-\left (x^{5}+11 x^{4}+42 x^{3}+54 x^{2}-27 x -81\right ) x^{3} {\mathrm e}^{-16} {\mathrm e}^{-\frac {16}{3}}\)	\(37\)
default	\({\mathrm e}^{-\frac {16}{3}} {\mathrm e}^{-16} \left (-x^{8}-11 x^{7}-42 x^{6}-54 x^{5}+27 x^{4}+81 x^{3}\right )\)	\(41\)
risch	\(-{\mathrm e}^{-\frac {64}{3}} x^{8}-11 \,{\mathrm e}^{-\frac {64}{3}} x^{7}-42 \,{\mathrm e}^{-\frac {64}{3}} x^{6}-54 \,{\mathrm e}^{-\frac {64}{3}} x^{5}+27 \,{\mathrm e}^{-\frac {64}{3}} x^{4}+81 \,{\mathrm e}^{-\frac {64}{3}} x^{3}\)	\(44\)
norman	\(\left (81 \,{\mathrm e}^{-4} {\mathrm e}^{-\frac {4}{3}} x^{3}+27 \,{\mathrm e}^{-4} {\mathrm e}^{-\frac {4}{3}} x^{4}-54 \,{\mathrm e}^{-4} {\mathrm e}^{-\frac {4}{3}} x^{5}-42 \,{\mathrm e}^{-4} {\mathrm e}^{-\frac {4}{3}} x^{6}-11 \,{\mathrm e}^{-4} {\mathrm e}^{-\frac {4}{3}} x^{7}-{\mathrm e}^{-4} {\mathrm e}^{-\frac {4}{3}} x^{8}\right ) {\mathrm e}^{-12} {\mathrm e}^{-4}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^7-77*x^6-252*x^5-270*x^4+108*x^3+243*x^2)/exp(4/3)^4/exp(4)^4,x,method=_RETURNVERBOSE)

[Out]

-(x^5+11*x^4+42*x^3+54*x^2-27*x-81)*x^3/exp(4)^4/exp(4/3)^4

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maxima [B]  time = 0.36, size = 33, normalized size = 1.74 \begin {gather*} -{\left (x^{8} + 11 \, x^{7} + 42 \, x^{6} + 54 \, x^{5} - 27 \, x^{4} - 81 \, x^{3}\right )} e^{\left (-\frac {64}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^7-77*x^6-252*x^5-270*x^4+108*x^3+243*x^2)/exp(4/3)^4/exp(4)^4,x, algorithm="maxima")

[Out]

-(x^8 + 11*x^7 + 42*x^6 + 54*x^5 - 27*x^4 - 81*x^3)*e^(-64/3)

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mupad [B]  time = 0.05, size = 43, normalized size = 2.26 \begin {gather*} -{\mathrm {e}}^{-\frac {64}{3}}\,x^8-11\,{\mathrm {e}}^{-\frac {64}{3}}\,x^7-42\,{\mathrm {e}}^{-\frac {64}{3}}\,x^6-54\,{\mathrm {e}}^{-\frac {64}{3}}\,x^5+27\,{\mathrm {e}}^{-\frac {64}{3}}\,x^4+81\,{\mathrm {e}}^{-\frac {64}{3}}\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-64/3)*(270*x^4 - 108*x^3 - 243*x^2 + 252*x^5 + 77*x^6 + 8*x^7),x)

[Out]

81*x^3*exp(-64/3) + 27*x^4*exp(-64/3) - 54*x^5*exp(-64/3) - 42*x^6*exp(-64/3) - 11*x^7*exp(-64/3) - x^8*exp(-6
4/3)

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sympy [B]  time = 0.06, size = 58, normalized size = 3.05 \begin {gather*} - \frac {x^{8}}{e^{\frac {64}{3}}} - \frac {11 x^{7}}{e^{\frac {64}{3}}} - \frac {42 x^{6}}{e^{\frac {64}{3}}} - \frac {54 x^{5}}{e^{\frac {64}{3}}} + \frac {27 x^{4}}{e^{\frac {64}{3}}} + \frac {81 x^{3}}{e^{\frac {64}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**7-77*x**6-252*x**5-270*x**4+108*x**3+243*x**2)/exp(4/3)**4/exp(4)**4,x)

[Out]

-x**8*exp(-64/3) - 11*x**7*exp(-64/3) - 42*x**6*exp(-64/3) - 54*x**5*exp(-64/3) + 27*x**4*exp(-64/3) + 81*x**3
*exp(-64/3)

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