3.30.42 \(\int \frac {e^{-x} (-20 e^x-1280 x^2+640 x^5-160 x^6+40 x^9-5 x^{10})}{x^2} \, dx\)

Optimal. Leaf size=25 \[ 5 \left (\frac {4+2 x}{x}+e^{-x} \left (16+x^4\right )^2\right ) \]

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Rubi [A]  time = 0.92, antiderivative size = 33, normalized size of antiderivative = 1.32, number of steps used = 31, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6741, 12, 6742, 2194, 2176} \begin {gather*} 5 e^{-x} x^8+160 e^{-x} x^4+1280 e^{-x}+\frac {20}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20*E^x - 1280*x^2 + 640*x^5 - 160*x^6 + 40*x^9 - 5*x^10)/(E^x*x^2),x]

[Out]

1280/E^x + 20/x + (160*x^4)/E^x + (5*x^8)/E^x

Rule 12

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{-x} \left (-4 e^x-256 x^2+128 x^5-32 x^6+8 x^9-x^{10}\right )}{x^2} \, dx\\ &=5 \int \frac {e^{-x} \left (-4 e^x-256 x^2+128 x^5-32 x^6+8 x^9-x^{10}\right )}{x^2} \, dx\\ &=5 \int \left (-256 e^{-x}-\frac {4}{x^2}+128 e^{-x} x^3-32 e^{-x} x^4+8 e^{-x} x^7-e^{-x} x^8\right ) \, dx\\ &=\frac {20}{x}-5 \int e^{-x} x^8 \, dx+40 \int e^{-x} x^7 \, dx-160 \int e^{-x} x^4 \, dx+640 \int e^{-x} x^3 \, dx-1280 \int e^{-x} \, dx\\ &=1280 e^{-x}+\frac {20}{x}-640 e^{-x} x^3+160 e^{-x} x^4-40 e^{-x} x^7+5 e^{-x} x^8-40 \int e^{-x} x^7 \, dx+280 \int e^{-x} x^6 \, dx-640 \int e^{-x} x^3 \, dx+1920 \int e^{-x} x^2 \, dx\\ &=1280 e^{-x}+\frac {20}{x}-1920 e^{-x} x^2+160 e^{-x} x^4-280 e^{-x} x^6+5 e^{-x} x^8-280 \int e^{-x} x^6 \, dx+1680 \int e^{-x} x^5 \, dx-1920 \int e^{-x} x^2 \, dx+3840 \int e^{-x} x \, dx\\ &=1280 e^{-x}+\frac {20}{x}-3840 e^{-x} x+160 e^{-x} x^4-1680 e^{-x} x^5+5 e^{-x} x^8-1680 \int e^{-x} x^5 \, dx+3840 \int e^{-x} \, dx-3840 \int e^{-x} x \, dx+8400 \int e^{-x} x^4 \, dx\\ &=-2560 e^{-x}+\frac {20}{x}-8240 e^{-x} x^4+5 e^{-x} x^8-3840 \int e^{-x} \, dx-8400 \int e^{-x} x^4 \, dx+33600 \int e^{-x} x^3 \, dx\\ &=1280 e^{-x}+\frac {20}{x}-33600 e^{-x} x^3+160 e^{-x} x^4+5 e^{-x} x^8-33600 \int e^{-x} x^3 \, dx+100800 \int e^{-x} x^2 \, dx\\ &=1280 e^{-x}+\frac {20}{x}-100800 e^{-x} x^2+160 e^{-x} x^4+5 e^{-x} x^8-100800 \int e^{-x} x^2 \, dx+201600 \int e^{-x} x \, dx\\ &=1280 e^{-x}+\frac {20}{x}-201600 e^{-x} x+160 e^{-x} x^4+5 e^{-x} x^8+201600 \int e^{-x} \, dx-201600 \int e^{-x} x \, dx\\ &=-200320 e^{-x}+\frac {20}{x}+160 e^{-x} x^4+5 e^{-x} x^8-201600 \int e^{-x} \, dx\\ &=1280 e^{-x}+\frac {20}{x}+160 e^{-x} x^4+5 e^{-x} x^8\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} \frac {20}{x}-5 e^{-x} \left (-256-32 x^4-x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20*E^x - 1280*x^2 + 640*x^5 - 160*x^6 + 40*x^9 - 5*x^10)/(E^x*x^2),x]

[Out]

20/x - (5*(-256 - 32*x^4 - x^8))/E^x

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fricas [A]  time = 1.51, size = 25, normalized size = 1.00 \begin {gather*} \frac {5 \, {\left (x^{9} + 32 \, x^{5} + 256 \, x + 4 \, e^{x}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(x)-5*x^10+40*x^9-160*x^6+640*x^5-1280*x^2)/exp(x)/x^2,x, algorithm="fricas")

[Out]

5*(x^9 + 32*x^5 + 256*x + 4*e^x)*e^(-x)/x

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giac [A]  time = 0.21, size = 31, normalized size = 1.24 \begin {gather*} \frac {5 \, {\left (x^{9} e^{\left (-x\right )} + 32 \, x^{5} e^{\left (-x\right )} + 256 \, x e^{\left (-x\right )} + 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(x)-5*x^10+40*x^9-160*x^6+640*x^5-1280*x^2)/exp(x)/x^2,x, algorithm="giac")

[Out]

5*(x^9*e^(-x) + 32*x^5*e^(-x) + 256*x*e^(-x) + 4)/x

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maple [A]  time = 0.02, size = 24, normalized size = 0.96

method	result	size
risch	\(\frac {20}{x}+\left (5 x^{8}+160 x^{4}+1280\right ) {\mathrm e}^{-x}\)	\(24\)
norman	\(\frac {\left (1280 x +160 x^{5}+5 x^{9}+20 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x}\)	\(27\)
default	\(\frac {20}{x}+1280 \,{\mathrm e}^{-x}+160 x^{4} {\mathrm e}^{-x}+5 \,{\mathrm e}^{-x} x^{8}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-20*exp(x)-5*x^10+40*x^9-160*x^6+640*x^5-1280*x^2)/exp(x)/x^2,x,method=_RETURNVERBOSE)

[Out]

20/x+(5*x^8+160*x^4+1280)*exp(-x)

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maxima [B]  time = 0.44, size = 138, normalized size = 5.52 \begin {gather*} 5 \, {\left (x^{8} + 8 \, x^{7} + 56 \, x^{6} + 336 \, x^{5} + 1680 \, x^{4} + 6720 \, x^{3} + 20160 \, x^{2} + 40320 \, x + 40320\right )} e^{\left (-x\right )} - 40 \, {\left (x^{7} + 7 \, x^{6} + 42 \, x^{5} + 210 \, x^{4} + 840 \, x^{3} + 2520 \, x^{2} + 5040 \, x + 5040\right )} e^{\left (-x\right )} + 160 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} - 640 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} + \frac {20}{x} + 1280 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(x)-5*x^10+40*x^9-160*x^6+640*x^5-1280*x^2)/exp(x)/x^2,x, algorithm="maxima")

[Out]

5*(x^8 + 8*x^7 + 56*x^6 + 336*x^5 + 1680*x^4 + 6720*x^3 + 20160*x^2 + 40320*x + 40320)*e^(-x) - 40*(x^7 + 7*x^
6 + 42*x^5 + 210*x^4 + 840*x^3 + 2520*x^2 + 5040*x + 5040)*e^(-x) + 160*(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(
-x) - 640*(x^3 + 3*x^2 + 6*x + 6)*e^(-x) + 20/x + 1280*e^(-x)

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mupad [B]  time = 1.80, size = 19, normalized size = 0.76 \begin {gather*} 5\,{\mathrm {e}}^{-x}\,{\left (x^4+16\right )}^2+\frac {20}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(20*exp(x) + 1280*x^2 - 640*x^5 + 160*x^6 - 40*x^9 + 5*x^10))/x^2,x)

[Out]

5*exp(-x)*(x^4 + 16)^2 + 20/x

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sympy [A]  time = 0.12, size = 17, normalized size = 0.68 \begin {gather*} \left (5 x^{8} + 160 x^{4} + 1280\right ) e^{- x} + \frac {20}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(x)-5*x**10+40*x**9-160*x**6+640*x**5-1280*x**2)/exp(x)/x**2,x)

[Out]

(5*x**8 + 160*x**4 + 1280)*exp(-x) + 20/x

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