3.102.16 \(\int \frac {e^{\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}} (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5)}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx\)

Optimal. Leaf size=31 \[ e^{2+3 x+x^2-\left (-5-x+\frac {x^2}{256 (5+x)^2}\right )^2} \]

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Rubi [F]  time = 1.92, 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 {\exp \left (\frac {-942080000-1040384000 x-455411200 x^2-98920960 x^3-10674689 x^4-458240 x^5}{40960000+32768000 x+9830400 x^2+1310720 x^3+65536 x^4}\right ) \left (-358400000-358240000 x-143248000 x^2-28643205 x^3-2864000 x^4-114560 x^5\right )}{51200000+51200000 x+20480000 x^2+4096000 x^3+409600 x^4+16384 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-942080000 - 1040384000*x - 455411200*x^2 - 98920960*x^3 - 10674689*x^4 - 458240*x^5)/(40960000 + 327
68000*x + 9830400*x^2 + 1310720*x^3 + 65536*x^4))*(-358400000 - 358240000*x - 143248000*x^2 - 28643205*x^3 - 2
864000*x^4 - 114560*x^5))/(51200000 + 51200000*x + 20480000*x^2 + 4096000*x^3 + 409600*x^4 + 16384*x^5),x]

[Out]

(-895*Defer[Int][E^(-1/65536*(942080000 + 1040384000*x + 455411200*x^2 + 98920960*x^3 + 10674689*x^4 + 458240*
x^5)/(5 + x)^4), x])/128 + (625*Defer[Int][1/(E^((942080000 + 1040384000*x + 455411200*x^2 + 98920960*x^3 + 10
674689*x^4 + 458240*x^5)/(65536*(5 + x)^4))*(5 + x)^5), x])/16384 - (375*Defer[Int][1/(E^((942080000 + 1040384
000*x + 455411200*x^2 + 98920960*x^3 + 10674689*x^4 + 458240*x^5)/(65536*(5 + x)^4))*(5 + x)^4), x])/16384 + (
75*Defer[Int][1/(E^((942080000 + 1040384000*x + 455411200*x^2 + 98920960*x^3 + 10674689*x^4 + 458240*x^5)/(655
36*(5 + x)^4))*(5 + x)^3), x])/16384 - (3205*Defer[Int][1/(E^((942080000 + 1040384000*x + 455411200*x^2 + 9892
0960*x^3 + 10674689*x^4 + 458240*x^5)/(65536*(5 + x)^4))*(5 + x)^2), x])/16384

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right ) \left (-71680000-71648000 x-28649600 x^2-5728641 x^3-572800 x^4-22912 x^5\right )}{16384 (5+x)^5} \, dx\\ &=\frac {5 \int \frac {\exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right ) \left (-71680000-71648000 x-28649600 x^2-5728641 x^3-572800 x^4-22912 x^5\right )}{(5+x)^5} \, dx}{16384}\\ &=\frac {5 \int \left (-22912 \exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )+\frac {125 \exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^5}-\frac {75 \exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^4}+\frac {15 \exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^3}-\frac {641 \exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^2}\right ) \, dx}{16384}\\ &=\frac {75 \int \frac {\exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^3} \, dx}{16384}-\frac {375 \int \frac {\exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^4} \, dx}{16384}+\frac {625 \int \frac {\exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^5} \, dx}{16384}-\frac {3205 \int \frac {\exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right )}{(5+x)^2} \, dx}{16384}-\frac {895}{128} \int \exp \left (-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.41, size = 36, normalized size = 1.16 \begin {gather*} e^{-\frac {942080000+1040384000 x+455411200 x^2+98920960 x^3+10674689 x^4+458240 x^5}{65536 (5+x)^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-942080000 - 1040384000*x - 455411200*x^2 - 98920960*x^3 - 10674689*x^4 - 458240*x^5)/(40960000
+ 32768000*x + 9830400*x^2 + 1310720*x^3 + 65536*x^4))*(-358400000 - 358240000*x - 143248000*x^2 - 28643205*x
^3 - 2864000*x^4 - 114560*x^5))/(51200000 + 51200000*x + 20480000*x^2 + 4096000*x^3 + 409600*x^4 + 16384*x^5),
x]

[Out]

E^(-1/65536*(942080000 + 1040384000*x + 455411200*x^2 + 98920960*x^3 + 10674689*x^4 + 458240*x^5)/(5 + x)^4)

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fricas [A]  time = 1.15, size = 48, normalized size = 1.55 \begin {gather*} e^{\left (-\frac {458240 \, x^{5} + 10674689 \, x^{4} + 98920960 \, x^{3} + 455411200 \, x^{2} + 1040384000 \, x + 942080000}{65536 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x-358400000)*exp((-458240*x^5-10674689
*x^4-98920960*x^3-455411200*x^2-1040384000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000
))/(16384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x, algorithm="fricas")

[Out]

e^(-1/65536*(458240*x^5 + 10674689*x^4 + 98920960*x^3 + 455411200*x^2 + 1040384000*x + 942080000)/(x^4 + 20*x^
3 + 150*x^2 + 500*x + 625))

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giac [B]  time = 0.18, size = 147, normalized size = 4.74 \begin {gather*} e^{\left (-\frac {895 \, x^{5}}{128 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {10674689 \, x^{4}}{65536 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {193205 \, x^{3}}{128 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {889475 \, x^{2}}{128 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} - \frac {15875 \, x}{x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625} - \frac {14375}{x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x-358400000)*exp((-458240*x^5-10674689
*x^4-98920960*x^3-455411200*x^2-1040384000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000
))/(16384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x, algorithm="giac")

[Out]

e^(-895/128*x^5/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 10674689/65536*x^4/(x^4 + 20*x^3 + 150*x^2 + 500*x +
625) - 193205/128*x^3/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 889475/128*x^2/(x^4 + 20*x^3 + 150*x^2 + 500*x
+ 625) - 15875*x/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) - 14375/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625))

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 hanged

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x-358400000)*exp((-458240*x^5-10674689*x^4-9
8920960*x^3-455411200*x^2-1040384000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000))/(16
384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x,method=_RETURNVERBOSE)

[Out]

int((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x-358400000)*exp((-458240*x^5-10674689*x^4-9
8920960*x^3-455411200*x^2-1040384000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000))/(16
384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x,method=_RETURNVERBOSE)

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maxima [B]  time = 3.30, size = 64, normalized size = 2.06 \begin {gather*} e^{\left (-\frac {895}{128} \, x - \frac {625}{65536 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )}} + \frac {125}{16384 \, {\left (x^{3} + 15 \, x^{2} + 75 \, x + 125\right )}} - \frac {75}{32768 \, {\left (x^{2} + 10 \, x + 25\right )}} + \frac {3205}{16384 \, {\left (x + 5\right )}} - \frac {1509889}{65536}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114560*x^5-2864000*x^4-28643205*x^3-143248000*x^2-358240000*x-358400000)*exp((-458240*x^5-10674689
*x^4-98920960*x^3-455411200*x^2-1040384000*x-942080000)/(65536*x^4+1310720*x^3+9830400*x^2+32768000*x+40960000
))/(16384*x^5+409600*x^4+4096000*x^3+20480000*x^2+51200000*x+51200000),x, algorithm="maxima")

[Out]

e^(-895/128*x - 625/65536/(x^4 + 20*x^3 + 150*x^2 + 500*x + 625) + 125/16384/(x^3 + 15*x^2 + 75*x + 125) - 75/
32768/(x^2 + 10*x + 25) + 3205/16384/(x + 5) - 1509889/65536)

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mupad [B]  time = 7.42, size = 160, normalized size = 5.16 \begin {gather*} {\mathrm {e}}^{-\frac {15875\,x}{x^4+20\,x^3+150\,x^2+500\,x+625}}\,{\mathrm {e}}^{-\frac {14375}{x^4+20\,x^3+150\,x^2+500\,x+625}}\,{\mathrm {e}}^{-\frac {895\,x^5}{128\,x^4+2560\,x^3+19200\,x^2+64000\,x+80000}}\,{\mathrm {e}}^{-\frac {193205\,x^3}{128\,x^4+2560\,x^3+19200\,x^2+64000\,x+80000}}\,{\mathrm {e}}^{-\frac {889475\,x^2}{128\,x^4+2560\,x^3+19200\,x^2+64000\,x+80000}}\,{\mathrm {e}}^{-\frac {10674689\,x^4}{65536\,x^4+1310720\,x^3+9830400\,x^2+32768000\,x+40960000}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(1040384000*x + 455411200*x^2 + 98920960*x^3 + 10674689*x^4 + 458240*x^5 + 942080000)/(32768000*x +
9830400*x^2 + 1310720*x^3 + 65536*x^4 + 40960000))*(358240000*x + 143248000*x^2 + 28643205*x^3 + 2864000*x^4
+ 114560*x^5 + 358400000))/(51200000*x + 20480000*x^2 + 4096000*x^3 + 409600*x^4 + 16384*x^5 + 51200000),x)

[Out]

exp(-(15875*x)/(500*x + 150*x^2 + 20*x^3 + x^4 + 625))*exp(-14375/(500*x + 150*x^2 + 20*x^3 + x^4 + 625))*exp(
-(895*x^5)/(64000*x + 19200*x^2 + 2560*x^3 + 128*x^4 + 80000))*exp(-(193205*x^3)/(64000*x + 19200*x^2 + 2560*x
^3 + 128*x^4 + 80000))*exp(-(889475*x^2)/(64000*x + 19200*x^2 + 2560*x^3 + 128*x^4 + 80000))*exp(-(10674689*x^
4)/(32768000*x + 9830400*x^2 + 1310720*x^3 + 65536*x^4 + 40960000))

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sympy [A]  time = 0.40, size = 48, normalized size = 1.55 \begin {gather*} e^{\frac {- 458240 x^{5} - 10674689 x^{4} - 98920960 x^{3} - 455411200 x^{2} - 1040384000 x - 942080000}{65536 x^{4} + 1310720 x^{3} + 9830400 x^{2} + 32768000 x + 40960000}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114560*x**5-2864000*x**4-28643205*x**3-143248000*x**2-358240000*x-358400000)*exp((-458240*x**5-106
74689*x**4-98920960*x**3-455411200*x**2-1040384000*x-942080000)/(65536*x**4+1310720*x**3+9830400*x**2+32768000
*x+40960000))/(16384*x**5+409600*x**4+4096000*x**3+20480000*x**2+51200000*x+51200000),x)

[Out]

exp((-458240*x**5 - 10674689*x**4 - 98920960*x**3 - 455411200*x**2 - 1040384000*x - 942080000)/(65536*x**4 + 1
310720*x**3 + 9830400*x**2 + 32768000*x + 40960000))

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