3.88.27 \(\int \frac {e^{\frac {16 e^{-\frac {10}{-5+x}}-32 e^{-\frac {5}{-5+x}} x+16 x^2}{-75 x^7+25 x^8}} (6000 x^2-4800 x^3+1200 x^4-96 x^5+e^{-\frac {10}{-5+x}} (8400-7040 x+1776 x^2-128 x^3)+e^{-\frac {5}{-5+x}} (-14400 x+11840 x^2-2976 x^3+224 x^4))}{5625 x^8-6000 x^9+2350 x^{10}-400 x^{11}+25 x^{12}} \, dx\)

Optimal. Leaf size=31 \[ e^{\frac {16 \left (e^{\frac {5}{5-x}}-x\right )^2}{25 (-3+x) x^7}} \]

________________________________________________________________________________________

Rubi [F]  time = 38.52, 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 {16 e^{-\frac {10}{-5+x}}-32 e^{-\frac {5}{-5+x}} x+16 x^2}{-75 x^7+25 x^8}\right ) \left (6000 x^2-4800 x^3+1200 x^4-96 x^5+e^{-\frac {10}{-5+x}} \left (8400-7040 x+1776 x^2-128 x^3\right )+e^{-\frac {5}{-5+x}} \left (-14400 x+11840 x^2-2976 x^3+224 x^4\right )\right )}{5625 x^8-6000 x^9+2350 x^{10}-400 x^{11}+25 x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((16/E^(10/(-5 + x)) - (32*x)/E^(5/(-5 + x)) + 16*x^2)/(-75*x^7 + 25*x^8))*(6000*x^2 - 4800*x^3 + 1200*
x^4 - 96*x^5 + (8400 - 7040*x + 1776*x^2 - 128*x^3)/E^(10/(-5 + x)) + (-14400*x + 11840*x^2 - 2976*x^3 + 224*x
^4)/E^(5/(-5 + x))))/(5625*x^8 - 6000*x^9 + 2350*x^10 - 400*x^11 + 25*x^12),x]

[Out]

(16*Defer[Int][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/(-5 + x)^2,
x])/390625 - (16*Defer[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7)
)/(-5 + x)^2, x])/78125 - (152*Defer[Int][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))
*(-3 + x)*x^7))/(-5 + x), x])/1953125 + (136*Defer[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^
(10/(-5 + x))*(-3 + x)*x^7))/(-5 + x), x])/390625 - (16*Defer[Int][E^((16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10
/(-5 + x))*(-3 + x)*x^7))/(-3 + x)^2, x])/6075 - (16*Defer[Int][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^
2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/(-3 + x)^2, x])/54675 + (32*Defer[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(
-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/(-3 + x)^2, x])/18225 + (8*Defer[Int][E^(-10/(-5 + x) + (16*
(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/(-3 + x), x])/10935 - (8*Defer[Int][E^(-5/(-5 +
x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/(-3 + x), x])/3645 + (112*Defer[Int][E^
(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^8, x])/75 + (128*Defer[Int
][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^7, x])/375 - (64*Defer
[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^7, x])/25 + (16*Def
er[Int][E^((16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^6, x])/15 + (944*Defer[Int][E^(
-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^6, x])/16875 - (704*Defer[I
nt][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^6, x])/1125 + (64*Def
er[Int][E^((16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^5, x])/225 + (128*Defer[Int][E^
(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^5, x])/253125 - (2144*Defe
r[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^5, x])/16875 + (16
*Defer[Int][E^((16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^4, x])/225 - (2192*Defer[In
t][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^4, x])/421875 - (1376
*Defer[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^4, x])/84375
+ (32*Defer[Int][E^((16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^3, x])/2025 - (37504*D
efer[Int][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^3, x])/1139062
5 + (9728*Defer[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^3, x
])/3796875 + (16*Defer[Int][E^((16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^2, x])/6075
- (1327568*Defer[Int][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x^2
, x])/854296875 + (37504*Defer[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-5 + x))*(-3 +
x)*x^7))/x^2, x])/11390625 - (2792576*Defer[Int][E^(-10/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/(25*E^(10/(-
5 + x))*(-3 + x)*x^7))/x, x])/4271484375 + (525856*Defer[Int][E^(-5/(-5 + x) + (16*(-1 + E^(5/(-5 + x))*x)^2)/
(25*E^(10/(-5 + x))*(-3 + x)*x^7))/x, x])/284765625

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right ) \left (6000 x^2-4800 x^3+1200 x^4-96 x^5+e^{-\frac {10}{-5+x}} \left (8400-7040 x+1776 x^2-128 x^3\right )+e^{-\frac {5}{-5+x}} \left (-14400 x+11840 x^2-2976 x^3+224 x^4\right )\right )}{25 x^8 \left (15-8 x+x^2\right )^2} \, dx\\ &=\frac {1}{25} \int \frac {\exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right ) \left (6000 x^2-4800 x^3+1200 x^4-96 x^5+e^{-\frac {10}{-5+x}} \left (8400-7040 x+1776 x^2-128 x^3\right )+e^{-\frac {5}{-5+x}} \left (-14400 x+11840 x^2-2976 x^3+224 x^4\right )\right )}{x^8 \left (15-8 x+x^2\right )^2} \, dx\\ &=\frac {1}{25} \int \left (\frac {6000 \exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^6}-\frac {4800 \exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^5}+\frac {1200 \exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^4}-\frac {96 \exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^3}+\frac {32 \exp \left (-\frac {5}{-5+x}+\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right ) \left (-450+370 x-93 x^2+7 x^3\right )}{(-5+x)^2 (-3+x)^2 x^7}-\frac {16 \exp \left (-\frac {10}{-5+x}+\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right ) \left (-525+440 x-111 x^2+8 x^3\right )}{(-5+x)^2 (-3+x)^2 x^8}\right ) \, dx\\ &=-\left (\frac {16}{25} \int \frac {\exp \left (-\frac {10}{-5+x}+\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right ) \left (-525+440 x-111 x^2+8 x^3\right )}{(-5+x)^2 (-3+x)^2 x^8} \, dx\right )+\frac {32}{25} \int \frac {\exp \left (-\frac {5}{-5+x}+\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right ) \left (-450+370 x-93 x^2+7 x^3\right )}{(-5+x)^2 (-3+x)^2 x^7} \, dx-\frac {96}{25} \int \frac {\exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^3} \, dx+48 \int \frac {\exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^4} \, dx-192 \int \frac {\exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^5} \, dx+240 \int \frac {\exp \left (\frac {16 e^{-\frac {10}{-5+x}} \left (-1+e^{\frac {5}{-5+x}} x\right )^2}{25 (-3+x) x^7}\right )}{(-5+x)^2 (-3+x)^2 x^6} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.36, size = 29, normalized size = 0.94 \begin {gather*} e^{\frac {16 \left (e^{-\frac {5}{-5+x}}-x\right )^2}{25 (-3+x) x^7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((16/E^(10/(-5 + x)) - (32*x)/E^(5/(-5 + x)) + 16*x^2)/(-75*x^7 + 25*x^8))*(6000*x^2 - 4800*x^3 +
1200*x^4 - 96*x^5 + (8400 - 7040*x + 1776*x^2 - 128*x^3)/E^(10/(-5 + x)) + (-14400*x + 11840*x^2 - 2976*x^3 +
224*x^4)/E^(5/(-5 + x))))/(5625*x^8 - 6000*x^9 + 2350*x^10 - 400*x^11 + 25*x^12),x]

[Out]

E^((16*(E^(-5/(-5 + x)) - x)^2)/(25*(-3 + x)*x^7))

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 37, normalized size = 1.19 \begin {gather*} e^{\left (\frac {16 \, {\left (x^{2} - 2 \, x e^{\left (-\frac {5}{x - 5}\right )} + e^{\left (-\frac {10}{x - 5}\right )}\right )}}{25 \, {\left (x^{8} - 3 \, x^{7}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-128*x^3+1776*x^2-7040*x+8400)*exp(-5/(x-5))^2+(224*x^4-2976*x^3+11840*x^2-14400*x)*exp(-5/(x-5))-
96*x^5+1200*x^4-4800*x^3+6000*x^2)*exp((16*exp(-5/(x-5))^2-32*x*exp(-5/(x-5))+16*x^2)/(25*x^8-75*x^7))/(25*x^1
2-400*x^11+2350*x^10-6000*x^9+5625*x^8),x, algorithm="fricas")

[Out]

e^(16/25*(x^2 - 2*x*e^(-5/(x - 5)) + e^(-10/(x - 5)))/(x^8 - 3*x^7))

________________________________________________________________________________________

giac [B]  time = 1.18, size = 61, normalized size = 1.97 \begin {gather*} e^{\left (\frac {16 \, x^{2}}{25 \, {\left (x^{8} - 3 \, x^{7}\right )}} - \frac {32 \, x e^{\left (-\frac {5}{x - 5}\right )}}{25 \, {\left (x^{8} - 3 \, x^{7}\right )}} + \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{25 \, {\left (x^{8} - 3 \, x^{7}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-128*x^3+1776*x^2-7040*x+8400)*exp(-5/(x-5))^2+(224*x^4-2976*x^3+11840*x^2-14400*x)*exp(-5/(x-5))-
96*x^5+1200*x^4-4800*x^3+6000*x^2)*exp((16*exp(-5/(x-5))^2-32*x*exp(-5/(x-5))+16*x^2)/(25*x^8-75*x^7))/(25*x^1
2-400*x^11+2350*x^10-6000*x^9+5625*x^8),x, algorithm="giac")

[Out]

e^(16/25*x^2/(x^8 - 3*x^7) - 32/25*x*e^(-5/(x - 5))/(x^8 - 3*x^7) + 16/25*e^(-10/(x - 5))/(x^8 - 3*x^7))

________________________________________________________________________________________

maple [A]  time = 0.12, size = 35, normalized size = 1.13

method	result	size
risch	\({\mathrm e}^{\frac {-\frac {32 x \,{\mathrm e}^{-\frac {5}{x -5}}}{25}+\frac {16 x^{2}}{25}+\frac {16 \,{\mathrm e}^{-\frac {10}{x -5}}}{25}}{x^{7} \left (x -3\right )}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-128*x^3+1776*x^2-7040*x+8400)*exp(-5/(x-5))^2+(224*x^4-2976*x^3+11840*x^2-14400*x)*exp(-5/(x-5))-96*x^5
+1200*x^4-4800*x^3+6000*x^2)*exp((16*exp(-5/(x-5))^2-32*x*exp(-5/(x-5))+16*x^2)/(25*x^8-75*x^7))/(25*x^12-400*
x^11+2350*x^10-6000*x^9+5625*x^8),x,method=_RETURNVERBOSE)

[Out]

exp(16/25*(-2*x*exp(-5/(x-5))+x^2+exp(-10/(x-5)))/x^7/(x-3))

________________________________________________________________________________________

maxima [B]  time = 1.16, size = 233, normalized size = 7.52 \begin {gather*} e^{\left (-\frac {32 \, e^{\left (-\frac {5}{x - 5}\right )}}{18225 \, {\left (x - 3\right )}} + \frac {32 \, e^{\left (-\frac {5}{x - 5}\right )}}{18225 \, x} + \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{54675 \, {\left (x - 3\right )}} - \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{54675 \, x} + \frac {16}{6075 \, {\left (x - 3\right )}} - \frac {16}{6075 \, x} + \frac {32 \, e^{\left (-\frac {5}{x - 5}\right )}}{6075 \, x^{2}} - \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{18225 \, x^{2}} - \frac {16}{2025 \, x^{2}} + \frac {32 \, e^{\left (-\frac {5}{x - 5}\right )}}{2025 \, x^{3}} - \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{6075 \, x^{3}} - \frac {16}{675 \, x^{3}} + \frac {32 \, e^{\left (-\frac {5}{x - 5}\right )}}{675 \, x^{4}} - \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{2025 \, x^{4}} - \frac {16}{225 \, x^{4}} + \frac {32 \, e^{\left (-\frac {5}{x - 5}\right )}}{225 \, x^{5}} - \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{675 \, x^{5}} - \frac {16}{75 \, x^{5}} + \frac {32 \, e^{\left (-\frac {5}{x - 5}\right )}}{75 \, x^{6}} - \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{225 \, x^{6}} - \frac {16 \, e^{\left (-\frac {10}{x - 5}\right )}}{75 \, x^{7}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-128*x^3+1776*x^2-7040*x+8400)*exp(-5/(x-5))^2+(224*x^4-2976*x^3+11840*x^2-14400*x)*exp(-5/(x-5))-
96*x^5+1200*x^4-4800*x^3+6000*x^2)*exp((16*exp(-5/(x-5))^2-32*x*exp(-5/(x-5))+16*x^2)/(25*x^8-75*x^7))/(25*x^1
2-400*x^11+2350*x^10-6000*x^9+5625*x^8),x, algorithm="maxima")

[Out]

e^(-32/18225*e^(-5/(x - 5))/(x - 3) + 32/18225*e^(-5/(x - 5))/x + 16/54675*e^(-10/(x - 5))/(x - 3) - 16/54675*
e^(-10/(x - 5))/x + 16/6075/(x - 3) - 16/6075/x + 32/6075*e^(-5/(x - 5))/x^2 - 16/18225*e^(-10/(x - 5))/x^2 -
16/2025/x^2 + 32/2025*e^(-5/(x - 5))/x^3 - 16/6075*e^(-10/(x - 5))/x^3 - 16/675/x^3 + 32/675*e^(-5/(x - 5))/x^
4 - 16/2025*e^(-10/(x - 5))/x^4 - 16/225/x^4 + 32/225*e^(-5/(x - 5))/x^5 - 16/675*e^(-10/(x - 5))/x^5 - 16/75/
x^5 + 32/75*e^(-5/(x - 5))/x^6 - 16/225*e^(-10/(x - 5))/x^6 - 16/75*e^(-10/(x - 5))/x^7)

________________________________________________________________________________________

mupad [B]  time = 6.03, size = 69, normalized size = 2.23 \begin {gather*} {\mathrm {e}}^{\frac {32\,x\,{\mathrm {e}}^{-\frac {5}{x-5}}}{75\,x^7-25\,x^8}}\,{\mathrm {e}}^{-\frac {16\,x^2}{75\,x^7-25\,x^8}}\,{\mathrm {e}}^{-\frac {16\,{\mathrm {e}}^{-\frac {10}{x-5}}}{75\,x^7-25\,x^8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(16*exp(-10/(x - 5)) - 32*x*exp(-5/(x - 5)) + 16*x^2)/(75*x^7 - 25*x^8))*(exp(-5/(x - 5))*(14400*x
- 11840*x^2 + 2976*x^3 - 224*x^4) - 6000*x^2 + 4800*x^3 - 1200*x^4 + 96*x^5 + exp(-10/(x - 5))*(7040*x - 1776*
x^2 + 128*x^3 - 8400)))/(5625*x^8 - 6000*x^9 + 2350*x^10 - 400*x^11 + 25*x^12),x)

[Out]

exp((32*x*exp(-5/(x - 5)))/(75*x^7 - 25*x^8))*exp(-(16*x^2)/(75*x^7 - 25*x^8))*exp(-(16*exp(-10/(x - 5)))/(75*
x^7 - 25*x^8))

________________________________________________________________________________________

sympy [A]  time = 0.93, size = 34, normalized size = 1.10 \begin {gather*} e^{\frac {16 x^{2} - 32 x e^{- \frac {5}{x - 5}} + 16 e^{- \frac {10}{x - 5}}}{25 x^{8} - 75 x^{7}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-128*x**3+1776*x**2-7040*x+8400)*exp(-5/(x-5))**2+(224*x**4-2976*x**3+11840*x**2-14400*x)*exp(-5/(
x-5))-96*x**5+1200*x**4-4800*x**3+6000*x**2)*exp((16*exp(-5/(x-5))**2-32*x*exp(-5/(x-5))+16*x**2)/(25*x**8-75*
x**7))/(25*x**12-400*x**11+2350*x**10-6000*x**9+5625*x**8),x)

[Out]

exp((16*x**2 - 32*x*exp(-5/(x - 5)) + 16*exp(-10/(x - 5)))/(25*x**8 - 75*x**7))

________________________________________________________________________________________