3.63.97 \(\int e^{-\frac {5}{x}+e^{-5/x} (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x (-2592 x^3-640 x^4-32 x^5))} (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} (80 x+48 x^2+32 x^3)+e^x (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5)) \, dx\)

Optimal. Leaf size=30 \[ e^{16 e^{-5/x} x^3 \left (-e^x+2 x+(9+x)^2\right )^2} \]

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Rubi [F]  time = 29.59, 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 \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) \left (524880 x+574128 x^2+252320 x^3+48160 x^4+3920 x^5+112 x^6+e^{2 x} \left (80 x+48 x^2+32 x^3\right )+e^x \left (-12960 x-10976 x^2-5312 x^3-800 x^4-32 x^5\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^
4 - 32*x^5))/E^(5/x))*(524880*x + 574128*x^2 + 252320*x^3 + 48160*x^4 + 3920*x^5 + 112*x^6 + E^(2*x)*(80*x + 4
8*x^2 + 32*x^3) + E^x*(-12960*x - 10976*x^2 - 5312*x^3 - 800*x^4 - 32*x^5)),x]

[Out]

524880*Defer[Int][E^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-259
2*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x, x] - 12960*Defer[Int][E^(-5/x + x + (104976*x^3 + 16*E^(2*x)*x^3 + 5184
0*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x, x] + 80*Defer[Int][E^(-5
/x + 2*x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 -
32*x^5))/E^(5/x))*x, x] + 574128*Defer[Int][E^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 6
40*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^2, x] - 10976*Defer[Int][E^(-5/x + x + (10497
6*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x)
)*x^2, x] + 48*Defer[Int][E^(-5/x + 2*x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x
^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^2, x] + 252320*Defer[Int][E^(-5/x + (104976*x^3 + 16*E^(2*
x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^3, x] - 5312
*Defer[Int][E^(-5/x + x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*
x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^3, x] + 32*Defer[Int][E^(-5/x + 2*x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840
*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^3, x] + 48160*Defer[Int][E
^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 3
2*x^5))/E^(5/x))*x^4, x] - 800*Defer[Int][E^(-5/x + x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 +
640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^4, x] + 3920*Defer[Int][E^(-5/x + (104976*x^
3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^
5, x] - 32*Defer[Int][E^(-5/x + x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E
^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^5, x] + 112*Defer[Int][E^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 +
51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 - 640*x^4 - 32*x^5))/E^(5/x))*x^6, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (524880 \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x+574128 \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^2+252320 \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^3+48160 \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^4+3920 \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^5+112 \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^6+16 \exp \left (-\frac {5}{x}+2 x+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x \left (5+3 x+2 x^2\right )-32 \exp \left (-\frac {5}{x}+x+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x \left (405+343 x+166 x^2+25 x^3+x^4\right )\right ) \, dx\\ &=16 \int \exp \left (-\frac {5}{x}+2 x+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x \left (5+3 x+2 x^2\right ) \, dx-32 \int \exp \left (-\frac {5}{x}+x+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x \left (405+343 x+166 x^2+25 x^3+x^4\right ) \, dx+112 \int \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^6 \, dx+3920 \int \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^5 \, dx+48160 \int \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^4 \, dx+252320 \int \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^3 \, dx+524880 \int \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x \, dx+574128 \int \exp \left (-\frac {5}{x}+e^{-5/x} \left (104976 x^3+16 e^{2 x} x^3+51840 x^4+8992 x^5+640 x^6+16 x^7+e^x \left (-2592 x^3-640 x^4-32 x^5\right )\right )\right ) x^2 \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 29, normalized size = 0.97 \begin {gather*} e^{16 e^{-5/x} x^3 \left (81-e^x+20 x+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-5/x + (104976*x^3 + 16*E^(2*x)*x^3 + 51840*x^4 + 8992*x^5 + 640*x^6 + 16*x^7 + E^x*(-2592*x^3 -
640*x^4 - 32*x^5))/E^(5/x))*(524880*x + 574128*x^2 + 252320*x^3 + 48160*x^4 + 3920*x^5 + 112*x^6 + E^(2*x)*(80
*x + 48*x^2 + 32*x^3) + E^x*(-12960*x - 10976*x^2 - 5312*x^3 - 800*x^4 - 32*x^5)),x]

[Out]

E^((16*x^3*(81 - E^x + 20*x + x^2)^2)/E^(5/x))

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fricas [B]  time = 0.73, size = 84, normalized size = 2.80 \begin {gather*} e^{\left (\frac {16 \, x^{4} e^{\left (2 \, x - \frac {5}{x}\right )} - 32 \, {\left (x^{6} + 20 \, x^{5} + 81 \, x^{4}\right )} e^{\left (x - \frac {5}{x}\right )} + 16 \, {\left (x^{8} + 40 \, x^{7} + 562 \, x^{6} + 3240 \, x^{5} + 6561 \, x^{4}\right )} e^{\left (-\frac {5}{x}\right )} - 5}{x} + \frac {5}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x^2-12960*x)*exp(x)+112*x^6+3920*x^5+
48160*x^4+252320*x^3+574128*x^2+524880*x)*exp((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^
6+8992*x^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x, algorithm="fricas")

[Out]

e^((16*x^4*e^(2*x - 5/x) - 32*(x^6 + 20*x^5 + 81*x^4)*e^(x - 5/x) + 16*(x^8 + 40*x^7 + 562*x^6 + 3240*x^5 + 65
61*x^4)*e^(-5/x) - 5)/x + 5/x)

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giac [B]  time = 1.89, size = 234, normalized size = 7.80 \begin {gather*} e^{\left (\frac {{\left (16 \, x^{8} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} + 640 \, x^{7} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} - 32 \, x^{6} e^{\left (\frac {2 \, x^{2} - 5}{x} + \frac {x^{2} - 5}{x}\right )} + 8992 \, x^{6} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} - 640 \, x^{5} e^{\left (\frac {2 \, x^{2} - 5}{x} + \frac {x^{2} - 5}{x}\right )} + 51840 \, x^{5} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} + 16 \, x^{4} e^{\left (\frac {2 \, {\left (2 \, x^{2} - 5\right )}}{x}\right )} - 2592 \, x^{4} e^{\left (\frac {2 \, x^{2} - 5}{x} + \frac {x^{2} - 5}{x}\right )} + 104976 \, x^{4} e^{\left (\frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} - 5 \, e^{\left (\frac {2 \, x^{2} - 5}{x}\right )}\right )} e^{\left (-\frac {2 \, x^{2} - 5}{x}\right )}}{x} + \frac {2 \, x^{2} - 5}{x} - \frac {2 \, {\left (x^{2} - 5\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x^2-12960*x)*exp(x)+112*x^6+3920*x^5+
48160*x^4+252320*x^3+574128*x^2+524880*x)*exp((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^
6+8992*x^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x, algorithm="giac")

[Out]

e^((16*x^8*e^(2*(x^2 - 5)/x) + 640*x^7*e^(2*(x^2 - 5)/x) - 32*x^6*e^((2*x^2 - 5)/x + (x^2 - 5)/x) + 8992*x^6*e
^(2*(x^2 - 5)/x) - 640*x^5*e^((2*x^2 - 5)/x + (x^2 - 5)/x) + 51840*x^5*e^(2*(x^2 - 5)/x) + 16*x^4*e^(2*(2*x^2
- 5)/x) - 2592*x^4*e^((2*x^2 - 5)/x + (x^2 - 5)/x) + 104976*x^4*e^(2*(x^2 - 5)/x) - 5*e^((2*x^2 - 5)/x))*e^(-(
2*x^2 - 5)/x)/x + (2*x^2 - 5)/x - 2*(x^2 - 5)/x)

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maple [A]  time = 238.70, size = 55, normalized size = 1.83

method	result	size
risch	\({\mathrm e}^{-16 x^{3} \left (-x^{4}+2 \,{\mathrm e}^{x} x^{2}-40 x^{3}+40 \,{\mathrm e}^{x} x -562 x^{2}+162 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-3240 x -6561\right ) {\mathrm e}^{-\frac {5}{x}}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x^2-12960*x)*exp(x)+112*x^6+3920*x^5+48160*
x^4+252320*x^3+574128*x^2+524880*x)*exp((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^6+8992
*x^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x,method=_RETURNVERBOSE)

[Out]

exp(-16*x^3*(-x^4+2*exp(x)*x^2-40*x^3+40*exp(x)*x-562*x^2+162*exp(x)-exp(2*x)-3240*x-6561)*exp(-5/x))

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maxima [B]  time = 9.77, size = 111, normalized size = 3.70 \begin {gather*} e^{\left (16 \, x^{7} e^{\left (-\frac {5}{x}\right )} + 640 \, x^{6} e^{\left (-\frac {5}{x}\right )} - 32 \, x^{5} e^{\left (x - \frac {5}{x}\right )} + 8992 \, x^{5} e^{\left (-\frac {5}{x}\right )} - 640 \, x^{4} e^{\left (x - \frac {5}{x}\right )} + 51840 \, x^{4} e^{\left (-\frac {5}{x}\right )} + 16 \, x^{3} e^{\left (2 \, x - \frac {5}{x}\right )} - 2592 \, x^{3} e^{\left (x - \frac {5}{x}\right )} + 104976 \, x^{3} e^{\left (-\frac {5}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^3+48*x^2+80*x)*exp(x)^2+(-32*x^5-800*x^4-5312*x^3-10976*x^2-12960*x)*exp(x)+112*x^6+3920*x^5+
48160*x^4+252320*x^3+574128*x^2+524880*x)*exp((16*exp(x)^2*x^3+(-32*x^5-640*x^4-2592*x^3)*exp(x)+16*x^7+640*x^
6+8992*x^5+51840*x^4+104976*x^3)/exp(5/x))/exp(5/x),x, algorithm="maxima")

[Out]

e^(16*x^7*e^(-5/x) + 640*x^6*e^(-5/x) - 32*x^5*e^(x - 5/x) + 8992*x^5*e^(-5/x) - 640*x^4*e^(x - 5/x) + 51840*x
^4*e^(-5/x) + 16*x^3*e^(2*x - 5/x) - 2592*x^3*e^(x - 5/x) + 104976*x^3*e^(-5/x))

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mupad [B]  time = 4.45, size = 119, normalized size = 3.97 \begin {gather*} {\mathrm {e}}^{-32\,x^5\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-640\,x^4\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-2592\,x^3\,{\mathrm {e}}^{-\frac {5}{x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{16\,x^7\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{640\,x^6\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{8992\,x^5\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{51840\,x^4\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{104976\,x^3\,{\mathrm {e}}^{-\frac {5}{x}}}\,{\mathrm {e}}^{16\,x^3\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-\frac {5}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(-5/x)*(16*x^3*exp(2*x) - exp(x)*(2592*x^3 + 640*x^4 + 32*x^5) + 104976*x^3 + 51840*x^4 + 8992*x^5
+ 640*x^6 + 16*x^7))*exp(-5/x)*(524880*x + exp(2*x)*(80*x + 48*x^2 + 32*x^3) - exp(x)*(12960*x + 10976*x^2 + 5
312*x^3 + 800*x^4 + 32*x^5) + 574128*x^2 + 252320*x^3 + 48160*x^4 + 3920*x^5 + 112*x^6),x)

[Out]

exp(-32*x^5*exp(-5/x)*exp(x))*exp(-640*x^4*exp(-5/x)*exp(x))*exp(-2592*x^3*exp(-5/x)*exp(x))*exp(16*x^7*exp(-5
/x))*exp(640*x^6*exp(-5/x))*exp(8992*x^5*exp(-5/x))*exp(51840*x^4*exp(-5/x))*exp(104976*x^3*exp(-5/x))*exp(16*
x^3*exp(2*x)*exp(-5/x))

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sympy [B]  time = 11.10, size = 61, normalized size = 2.03 \begin {gather*} e^{\left (16 x^{7} + 640 x^{6} + 8992 x^{5} + 51840 x^{4} + 16 x^{3} e^{2 x} + 104976 x^{3} + \left (- 32 x^{5} - 640 x^{4} - 2592 x^{3}\right ) e^{x}\right ) e^{- \frac {5}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x**3+48*x**2+80*x)*exp(x)**2+(-32*x**5-800*x**4-5312*x**3-10976*x**2-12960*x)*exp(x)+112*x**6+3
920*x**5+48160*x**4+252320*x**3+574128*x**2+524880*x)*exp((16*exp(x)**2*x**3+(-32*x**5-640*x**4-2592*x**3)*exp
(x)+16*x**7+640*x**6+8992*x**5+51840*x**4+104976*x**3)/exp(5/x))/exp(5/x),x)

[Out]

exp((16*x**7 + 640*x**6 + 8992*x**5 + 51840*x**4 + 16*x**3*exp(2*x) + 104976*x**3 + (-32*x**5 - 640*x**4 - 259
2*x**3)*exp(x))*exp(-5/x))

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