3.31.5 \(\int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5)}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx\)

Optimal. Leaf size=21 \[ e^{-4-7 x-\frac {x^2}{(-4+x)^4}}-x \]

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Rubi [F]  time = 2.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+\exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}\right ) \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1024 - 1280*x + 640*x^2 - 160*x^3 + 20*x^4 - x^5 + E^((-1024 - 768*x + 1407*x^2 - 608*x^3 + 108*x^4 - 7*x
^5)/(256 - 256*x + 96*x^2 - 16*x^3 + x^4))*(7168 - 8952*x + 4482*x^2 - 1120*x^3 + 140*x^4 - 7*x^5))/(-1024 + 1
280*x - 640*x^2 + 160*x^3 - 20*x^4 + x^5),x]

[Out]

-2560/(4 - x)^4 + 2560/(4 - x)^3 - 960/(4 - x)^2 + 160/(4 - x) - x + (10*x^4)/(4 - x)^4 - 7*Defer[Int][E^((-10
24 - 768*x + 1407*x^2 - 608*x^3 + 108*x^4 - 7*x^5)/(-4 + x)^4), x] + 64*Defer[Int][E^((-1024 - 768*x + 1407*x^
2 - 608*x^3 + 108*x^4 - 7*x^5)/(-4 + x)^4)/(-4 + x)^5, x] + 24*Defer[Int][E^((-1024 - 768*x + 1407*x^2 - 608*x
^3 + 108*x^4 - 7*x^5)/(-4 + x)^4)/(-4 + x)^4, x] + 2*Defer[Int][E^((-1024 - 768*x + 1407*x^2 - 608*x^3 + 108*x
^4 - 7*x^5)/(-4 + x)^4)/(-4 + x)^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5-\exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right ) \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{(4-x)^5} \, dx\\ &=\int \left (\frac {1024}{(-4+x)^5}-\frac {1280 x}{(-4+x)^5}+\frac {640 x^2}{(-4+x)^5}-\frac {160 x^3}{(-4+x)^5}+\frac {20 x^4}{(-4+x)^5}-\frac {x^5}{(-4+x)^5}+\frac {\exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right ) \left (-7168+8952 x-4482 x^2+1120 x^3-140 x^4+7 x^5\right )}{(4-x)^5}\right ) \, dx\\ &=-\frac {256}{(4-x)^4}+20 \int \frac {x^4}{(-4+x)^5} \, dx-160 \int \frac {x^3}{(-4+x)^5} \, dx+640 \int \frac {x^2}{(-4+x)^5} \, dx-1280 \int \frac {x}{(-4+x)^5} \, dx-\int \frac {x^5}{(-4+x)^5} \, dx+\int \frac {\exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right ) \left (-7168+8952 x-4482 x^2+1120 x^3-140 x^4+7 x^5\right )}{(4-x)^5} \, dx\\ &=-\frac {256}{(4-x)^4}+\frac {10 x^4}{(4-x)^4}+20 \int \left (\frac {256}{(-4+x)^5}+\frac {256}{(-4+x)^4}+\frac {96}{(-4+x)^3}+\frac {16}{(-4+x)^2}+\frac {1}{-4+x}\right ) \, dx+640 \int \left (\frac {16}{(-4+x)^5}+\frac {8}{(-4+x)^4}+\frac {1}{(-4+x)^3}\right ) \, dx-1280 \int \left (\frac {4}{(-4+x)^5}+\frac {1}{(-4+x)^4}\right ) \, dx+\int \left (-7 \exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right )+\frac {64 \exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right )}{(-4+x)^5}+\frac {24 \exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right )}{(-4+x)^4}+\frac {2 \exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right )}{(-4+x)^3}\right ) \, dx-\int \left (1+\frac {1024}{(-4+x)^5}+\frac {1280}{(-4+x)^4}+\frac {640}{(-4+x)^3}+\frac {160}{(-4+x)^2}+\frac {20}{-4+x}\right ) \, dx\\ &=-\frac {2560}{(4-x)^4}+\frac {2560}{(4-x)^3}-\frac {960}{(4-x)^2}+\frac {160}{4-x}-x+\frac {10 x^4}{(4-x)^4}+2 \int \frac {\exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right )}{(-4+x)^3} \, dx-7 \int \exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right ) \, dx+24 \int \frac {\exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right )}{(-4+x)^4} \, dx+64 \int \frac {\exp \left (\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{(-4+x)^4}\right )}{(-4+x)^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.99, size = 33, normalized size = 1.57 \begin {gather*} 4+e^{-4-\frac {16}{(-4+x)^4}-\frac {8}{(-4+x)^3}-\frac {1}{(-4+x)^2}-7 x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1024 - 1280*x + 640*x^2 - 160*x^3 + 20*x^4 - x^5 + E^((-1024 - 768*x + 1407*x^2 - 608*x^3 + 108*x^4
- 7*x^5)/(256 - 256*x + 96*x^2 - 16*x^3 + x^4))*(7168 - 8952*x + 4482*x^2 - 1120*x^3 + 140*x^4 - 7*x^5))/(-10
24 + 1280*x - 640*x^2 + 160*x^3 - 20*x^4 + x^5),x]

[Out]

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

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fricas [B]  time = 0.66, size = 52, normalized size = 2.48 \begin {gather*} -x + e^{\left (-\frac {7 \, x^{5} - 108 \, x^{4} + 608 \, x^{3} - 1407 \, x^{2} + 768 \, x + 1024}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108*x^4-608*x^3+1407*x^2-768*x-1024)/(x^
4-16*x^3+96*x^2-256*x+256))-x^5+20*x^4-160*x^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x
, algorithm="fricas")

[Out]

-x + e^(-(7*x^5 - 108*x^4 + 608*x^3 - 1407*x^2 + 768*x + 1024)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256))

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giac [B]  time = 1.02, size = 58, normalized size = 2.76 \begin {gather*} -{\left (x e^{4} - e^{\left (-\frac {7 \, x^{5} - 112 \, x^{4} + 672 \, x^{3} - 1791 \, x^{2} + 1792 \, x}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256}\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108*x^4-608*x^3+1407*x^2-768*x-1024)/(x^
4-16*x^3+96*x^2-256*x+256))-x^5+20*x^4-160*x^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x
, algorithm="giac")

[Out]

-(x*e^4 - e^(-(7*x^5 - 112*x^4 + 672*x^3 - 1791*x^2 + 1792*x)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256)))*e^(-4)

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maple [A]  time = 1.21, size = 38, normalized size = 1.81

method	result	size
risch	\(-x +{\mathrm e}^{-\frac {7 x^{5}-108 x^{4}+608 x^{3}-1407 x^{2}+768 x +1024}{\left (x -4\right )^{4}}}\)	\(38\)
norman	\(\frac {x^{4} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-1280 x^{2}+160 x^{3}+3840 x -x^{5}-256 x \,{\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}+96 x^{2} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-16 x^{3} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}+256 \,{\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-4096}{\left (x -4\right )^{4}}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108*x^4-608*x^3+1407*x^2-768*x-1024)/(x^4-16*x
^3+96*x^2-256*x+256))-x^5+20*x^4-160*x^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x,metho
d=_RETURNVERBOSE)

[Out]

-x+exp(-(7*x^5-108*x^4+608*x^3-1407*x^2+768*x+1024)/(x-4)^4)

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maxima [B]  time = 0.69, size = 249, normalized size = 11.86 \begin {gather*} -x + \frac {32 \, {\left (15 \, x^{3} - 150 \, x^{2} + 520 \, x - 616\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} - \frac {320 \, {\left (3 \, x^{3} - 27 \, x^{2} + 88 \, x - 100\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} + \frac {160 \, {\left (x^{3} - 6 \, x^{2} + 16 \, x - 16\right )}}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256} - \frac {320 \, {\left (3 \, x^{2} - 8 \, x + 8\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} + \frac {1280 \, {\left (x - 1\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} - \frac {256}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256} + e^{\left (-7 \, x - \frac {16}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256} - \frac {8}{x^{3} - 12 \, x^{2} + 48 \, x - 64} - \frac {1}{x^{2} - 8 \, x + 16} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108*x^4-608*x^3+1407*x^2-768*x-1024)/(x^
4-16*x^3+96*x^2-256*x+256))-x^5+20*x^4-160*x^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x
, algorithm="maxima")

[Out]

-x + 32/3*(15*x^3 - 150*x^2 + 520*x - 616)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) - 320/3*(3*x^3 - 27*x^2 + 88*
x - 100)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) + 160*(x^3 - 6*x^2 + 16*x - 16)/(x^4 - 16*x^3 + 96*x^2 - 256*x
+ 256) - 320/3*(3*x^2 - 8*x + 8)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) + 1280/3*(x - 1)/(x^4 - 16*x^3 + 96*x^2
- 256*x + 256) - 256/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) + e^(-7*x - 16/(x^4 - 16*x^3 + 96*x^2 - 256*x + 25
6) - 8/(x^3 - 12*x^2 + 48*x - 64) - 1/(x^2 - 8*x + 16) - 4)

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mupad [B]  time = 1.96, size = 156, normalized size = 7.43 \begin {gather*} {\mathrm {e}}^{-\frac {768\,x}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{-\frac {7\,x^5}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{\frac {108\,x^4}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{-\frac {608\,x^3}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{\frac {1407\,x^2}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{-\frac {1024}{x^4-16\,x^3+96\,x^2-256\,x+256}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1280*x + exp(-(768*x - 1407*x^2 + 608*x^3 - 108*x^4 + 7*x^5 + 1024)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256
))*(8952*x - 4482*x^2 + 1120*x^3 - 140*x^4 + 7*x^5 - 7168) - 640*x^2 + 160*x^3 - 20*x^4 + x^5 - 1024)/(1280*x
- 640*x^2 + 160*x^3 - 20*x^4 + x^5 - 1024),x)

[Out]

exp(-(768*x)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp(-(7*x^5)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp((1
08*x^4)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp(-(608*x^3)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp((1407
*x^2)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp(-1024/(96*x^2 - 256*x - 16*x^3 + x^4 + 256)) - x

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sympy [B]  time = 0.33, size = 46, normalized size = 2.19 \begin {gather*} - x + e^{\frac {- 7 x^{5} + 108 x^{4} - 608 x^{3} + 1407 x^{2} - 768 x - 1024}{x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x**5+140*x**4-1120*x**3+4482*x**2-8952*x+7168)*exp((-7*x**5+108*x**4-608*x**3+1407*x**2-768*x-1
024)/(x**4-16*x**3+96*x**2-256*x+256))-x**5+20*x**4-160*x**3+640*x**2-1280*x+1024)/(x**5-20*x**4+160*x**3-640*
x**2+1280*x-1024),x)

[Out]

-x + exp((-7*x**5 + 108*x**4 - 608*x**3 + 1407*x**2 - 768*x - 1024)/(x**4 - 16*x**3 + 96*x**2 - 256*x + 256))

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