3.4.60 \(\int \frac {-21233664-27869184 x-14846976 x^2-3519936 x^3+219996 x^5+57996 x^6+6804 x^7+324 x^8}{e^8 x^5} \, dx\)

Optimal. Leaf size=19 \[ \frac {81 \left (-x+(4+x)^2\right )^4}{e^8 x^4} \]

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Rubi [B]  time = 0.04, antiderivative size = 63, normalized size of antiderivative = 3.32, number of steps used = 3, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 14} \begin {gather*} \frac {81 x^4}{e^8}+\frac {5308416}{e^8 x^4}+\frac {2268 x^3}{e^8}+\frac {9289728}{e^8 x^3}+\frac {28998 x^2}{e^8}+\frac {7423488}{e^8 x^2}+\frac {219996 x}{e^8}+\frac {3519936}{e^8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-21233664 - 27869184*x - 14846976*x^2 - 3519936*x^3 + 219996*x^5 + 57996*x^6 + 6804*x^7 + 324*x^8)/(E^8*x
^5),x]

[Out]

5308416/(E^8*x^4) + 9289728/(E^8*x^3) + 7423488/(E^8*x^2) + 3519936/(E^8*x) + (219996*x)/E^8 + (28998*x^2)/E^8
+ (2268*x^3)/E^8 + (81*x^4)/E^8

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-21233664-27869184 x-14846976 x^2-3519936 x^3+219996 x^5+57996 x^6+6804 x^7+324 x^8}{x^5} \, dx}{e^8}\\ &=\frac {\int \left (219996-\frac {21233664}{x^5}-\frac {27869184}{x^4}-\frac {14846976}{x^3}-\frac {3519936}{x^2}+57996 x+6804 x^2+324 x^3\right ) \, dx}{e^8}\\ &=\frac {5308416}{e^8 x^4}+\frac {9289728}{e^8 x^3}+\frac {7423488}{e^8 x^2}+\frac {3519936}{e^8 x}+\frac {219996 x}{e^8}+\frac {28998 x^2}{e^8}+\frac {2268 x^3}{e^8}+\frac {81 x^4}{e^8}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 48, normalized size = 2.53 \begin {gather*} \frac {324 \left (\frac {16384}{x^4}+\frac {28672}{x^3}+\frac {22912}{x^2}+\frac {10864}{x}+679 x+\frac {179 x^2}{2}+7 x^3+\frac {x^4}{4}\right )}{e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-21233664 - 27869184*x - 14846976*x^2 - 3519936*x^3 + 219996*x^5 + 57996*x^6 + 6804*x^7 + 324*x^8)/
(E^8*x^5),x]

[Out]

(324*(16384/x^4 + 28672/x^3 + 22912/x^2 + 10864/x + 679*x + (179*x^2)/2 + 7*x^3 + x^4/4))/E^8

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fricas [B]  time = 1.13, size = 40, normalized size = 2.11 \begin {gather*} \frac {81 \, {\left (x^{8} + 28 \, x^{7} + 358 \, x^{6} + 2716 \, x^{5} + 43456 \, x^{3} + 91648 \, x^{2} + 114688 \, x + 65536\right )} e^{\left (-8\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((324*x^8+6804*x^7+57996*x^6+219996*x^5-3519936*x^3-14846976*x^2-27869184*x-21233664)/x^5/exp(1)^8,x,
algorithm="fricas")

[Out]

81*(x^8 + 28*x^7 + 358*x^6 + 2716*x^5 + 43456*x^3 + 91648*x^2 + 114688*x + 65536)*e^(-8)/x^4

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giac [B]  time = 0.34, size = 41, normalized size = 2.16 \begin {gather*} 81 \, {\left (x^{4} + 28 \, x^{3} + 358 \, x^{2} + 2716 \, x + \frac {64 \, {\left (679 \, x^{3} + 1432 \, x^{2} + 1792 \, x + 1024\right )}}{x^{4}}\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((324*x^8+6804*x^7+57996*x^6+219996*x^5-3519936*x^3-14846976*x^2-27869184*x-21233664)/x^5/exp(1)^8,x,
algorithm="giac")

[Out]

81*(x^4 + 28*x^3 + 358*x^2 + 2716*x + 64*(679*x^3 + 1432*x^2 + 1792*x + 1024)/x^4)*e^(-8)

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maple [B]  time = 0.06, size = 43, normalized size = 2.26

method	result	size
gosper	\(\frac {81 \left (x^{8}+28 x^{7}+358 x^{6}+2716 x^{5}+43456 x^{3}+91648 x^{2}+114688 x +65536\right ) {\mathrm e}^{-8}}{x^{4}}\)	\(43\)
default	\(324 \,{\mathrm e}^{-8} \left (\frac {x^{4}}{4}+7 x^{3}+\frac {179 x^{2}}{2}+679 x +\frac {22912}{x^{2}}+\frac {28672}{x^{3}}+\frac {10864}{x}+\frac {16384}{x^{4}}\right )\)	\(46\)
risch	\(81 \,{\mathrm e}^{-8} x^{4}+2268 \,{\mathrm e}^{-8} x^{3}+28998 \,{\mathrm e}^{-8} x^{2}+219996 \,{\mathrm e}^{-8} x +\frac {{\mathrm e}^{-8} \left (3519936 x^{3}+7423488 x^{2}+9289728 x +5308416\right )}{x^{4}}\)	\(49\)
norman	\(\frac {\left (5308416 \,{\mathrm e}^{-1}+9289728 \,{\mathrm e}^{-1} x +7423488 x^{2} {\mathrm e}^{-1}+3519936 \,{\mathrm e}^{-1} x^{3}+219996 \,{\mathrm e}^{-1} x^{5}+28998 \,{\mathrm e}^{-1} x^{6}+2268 \,{\mathrm e}^{-1} x^{7}+81 \,{\mathrm e}^{-1} x^{8}\right ) {\mathrm e}^{-7}}{x^{4}}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((324*x^8+6804*x^7+57996*x^6+219996*x^5-3519936*x^3-14846976*x^2-27869184*x-21233664)/x^5/exp(1)^8,x,method
=_RETURNVERBOSE)

[Out]

81*(x^8+28*x^7+358*x^6+2716*x^5+43456*x^3+91648*x^2+114688*x+65536)/exp(1)^8/x^4

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maxima [B]  time = 0.36, size = 41, normalized size = 2.16 \begin {gather*} 81 \, {\left (x^{4} + 28 \, x^{3} + 358 \, x^{2} + 2716 \, x + \frac {64 \, {\left (679 \, x^{3} + 1432 \, x^{2} + 1792 \, x + 1024\right )}}{x^{4}}\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((324*x^8+6804*x^7+57996*x^6+219996*x^5-3519936*x^3-14846976*x^2-27869184*x-21233664)/x^5/exp(1)^8,x,
algorithm="maxima")

[Out]

81*(x^4 + 28*x^3 + 358*x^2 + 2716*x + 64*(679*x^3 + 1432*x^2 + 1792*x + 1024)/x^4)*e^(-8)

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mupad [B]  time = 0.05, size = 48, normalized size = 2.53 \begin {gather*} 219996\,x\,{\mathrm {e}}^{-8}+28998\,x^2\,{\mathrm {e}}^{-8}+2268\,x^3\,{\mathrm {e}}^{-8}+81\,x^4\,{\mathrm {e}}^{-8}+\frac {{\mathrm {e}}^{-8}\,\left (3519936\,x^3+7423488\,x^2+9289728\,x+5308416\right )}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-8)*(27869184*x + 14846976*x^2 + 3519936*x^3 - 219996*x^5 - 57996*x^6 - 6804*x^7 - 324*x^8 + 2123366
4))/x^5,x)

[Out]

219996*x*exp(-8) + 28998*x^2*exp(-8) + 2268*x^3*exp(-8) + 81*x^4*exp(-8) + (exp(-8)*(9289728*x + 7423488*x^2 +
3519936*x^3 + 5308416))/x^4

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sympy [B]  time = 0.10, size = 39, normalized size = 2.05 \begin {gather*} \frac {81 x^{4} + 2268 x^{3} + 28998 x^{2} + 219996 x + \frac {3519936 x^{3} + 7423488 x^{2} + 9289728 x + 5308416}{x^{4}}}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((324*x**8+6804*x**7+57996*x**6+219996*x**5-3519936*x**3-14846976*x**2-27869184*x-21233664)/x**5/exp(
1)**8,x)

[Out]

(81*x**4 + 2268*x**3 + 28998*x**2 + 219996*x + (3519936*x**3 + 7423488*x**2 + 9289728*x + 5308416)/x**4)*exp(-
8)

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