3.63.2 \(\int \frac {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7)}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx\)

Optimal. Leaf size=29 \[ (1+5 x) \left (4+\frac {10 \left (-e^{\frac {16}{\left (-4+x+x^2\right )^2}}+x\right )}{x^2}\right ) \]

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Rubi [F]  time = 5.02, 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 {640 x-480 x^2-1640 x^3+1190 x^4+810 x^5-490 x^6-190 x^7+60 x^8+20 x^9+e^{\frac {16}{16-8 x-7 x^2+2 x^3+x^4}} \left (-1280-1920 x+5360 x^2+4540 x^3-1330 x^4-390 x^5+170 x^6+50 x^7\right )}{-64 x^3+48 x^4+36 x^5-23 x^6-9 x^7+3 x^8+x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(640*x - 480*x^2 - 1640*x^3 + 1190*x^4 + 810*x^5 - 490*x^6 - 190*x^7 + 60*x^8 + 20*x^9 + E^(16/(16 - 8*x -
7*x^2 + 2*x^3 + x^4))*(-1280 - 1920*x + 5360*x^2 + 4540*x^3 - 1330*x^4 - 390*x^5 + 170*x^6 + 50*x^7))/(-64*x^
3 + 48*x^4 + 36*x^5 - 23*x^6 - 9*x^7 + 3*x^8 + x^9),x]

[Out]

10/x + 20*x - (30880*Defer[Int][E^(16/(-4 + x + x^2)^2)/(-1 + Sqrt[17] - 2*x)^3, x])/(17*Sqrt[17]) - (2320*(17
- Sqrt[17])*Defer[Int][E^(16/(-4 + x + x^2)^2)/(-1 + Sqrt[17] - 2*x)^3, x])/289 - (57880*Defer[Int][E^(16/(-4
+ x + x^2)^2)/(-1 + Sqrt[17] - 2*x)^2, x])/289 + (300*(1 - Sqrt[17])*Defer[Int][E^(16/(-4 + x + x^2)^2)/(-1 +
Sqrt[17] - 2*x)^2, x])/17 + (1160*(3 - Sqrt[17])*Defer[Int][E^(16/(-4 + x + x^2)^2)/(-1 + Sqrt[17] - 2*x)^2,
x])/289 - (2900*Defer[Int][E^(16/(-4 + x + x^2)^2)/(-1 + Sqrt[17] - 2*x), x])/(17*Sqrt[17]) + 20*Defer[Int][E^
(16/(-4 + x + x^2)^2)/x^3, x] + 45*Defer[Int][E^(16/(-4 + x + x^2)^2)/x^2, x] - (155*Defer[Int][E^(16/(-4 + x
+ x^2)^2)/x, x])/4 + (5*(527 + 39*Sqrt[17])*Defer[Int][E^(16/(-4 + x + x^2)^2)/(1 - Sqrt[17] + 2*x), x])/68 -
(30880*Defer[Int][E^(16/(-4 + x + x^2)^2)/(1 + Sqrt[17] + 2*x)^3, x])/(17*Sqrt[17]) + (2320*(17 + Sqrt[17])*De
fer[Int][E^(16/(-4 + x + x^2)^2)/(1 + Sqrt[17] + 2*x)^3, x])/289 - (57880*Defer[Int][E^(16/(-4 + x + x^2)^2)/(
1 + Sqrt[17] + 2*x)^2, x])/289 + (300*(1 + Sqrt[17])*Defer[Int][E^(16/(-4 + x + x^2)^2)/(1 + Sqrt[17] + 2*x)^2
, x])/17 + (1160*(3 + Sqrt[17])*Defer[Int][E^(16/(-4 + x + x^2)^2)/(1 + Sqrt[17] + 2*x)^2, x])/289 - (2900*Def
er[Int][E^(16/(-4 + x + x^2)^2)/(1 + Sqrt[17] + 2*x), x])/(17*Sqrt[17]) + (5*(527 - 39*Sqrt[17])*Defer[Int][E^
(16/(-4 + x + x^2)^2)/(1 + Sqrt[17] + 2*x), x])/68

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 \left (x \left (-1+2 x^2\right )+\frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} \left (-128-192 x+536 x^2+454 x^3-133 x^4-39 x^5+17 x^6+5 x^7\right )}{\left (-4+x+x^2\right )^3}\right )}{x^3} \, dx\\ &=10 \int \frac {x \left (-1+2 x^2\right )+\frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} \left (-128-192 x+536 x^2+454 x^3-133 x^4-39 x^5+17 x^6+5 x^7\right )}{\left (-4+x+x^2\right )^3}}{x^3} \, dx\\ &=10 \int \left (\frac {-1+2 x^2}{x^2}+\frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} \left (-128-192 x+536 x^2+454 x^3-133 x^4-39 x^5+17 x^6+5 x^7\right )}{x^3 \left (-4+x+x^2\right )^3}\right ) \, dx\\ &=10 \int \frac {-1+2 x^2}{x^2} \, dx+10 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} \left (-128-192 x+536 x^2+454 x^3-133 x^4-39 x^5+17 x^6+5 x^7\right )}{x^3 \left (-4+x+x^2\right )^3} \, dx\\ &=10 \int \left (2-\frac {1}{x^2}\right ) \, dx+10 \int \left (\frac {2 e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x^3}+\frac {9 e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{2 x^2}-\frac {31 e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{8 x}+\frac {2 e^{\frac {16}{\left (-4+x+x^2\right )^2}} (193+29 x)}{\left (-4+x+x^2\right )^3}+\frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} (-17-15 x)}{\left (-4+x+x^2\right )^2}+\frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} (35+31 x)}{8 \left (-4+x+x^2\right )}\right ) \, dx\\ &=\frac {10}{x}+20 x+\frac {5}{4} \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} (35+31 x)}{-4+x+x^2} \, dx+10 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} (-17-15 x)}{\left (-4+x+x^2\right )^2} \, dx+20 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x^3} \, dx+20 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} (193+29 x)}{\left (-4+x+x^2\right )^3} \, dx-\frac {155}{4} \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x} \, dx+45 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x^2} \, dx\\ &=\frac {10}{x}+20 x+\frac {5}{4} \int \left (\frac {\left (31+\frac {39}{\sqrt {17}}\right ) e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{1-\sqrt {17}+2 x}+\frac {\left (31-\frac {39}{\sqrt {17}}\right ) e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{1+\sqrt {17}+2 x}\right ) \, dx+10 \int \left (-\frac {17 e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{\left (-4+x+x^2\right )^2}-\frac {15 e^{\frac {16}{\left (-4+x+x^2\right )^2}} x}{\left (-4+x+x^2\right )^2}\right ) \, dx+20 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x^3} \, dx+20 \int \left (\frac {193 e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{\left (-4+x+x^2\right )^3}+\frac {29 e^{\frac {16}{\left (-4+x+x^2\right )^2}} x}{\left (-4+x+x^2\right )^3}\right ) \, dx-\frac {155}{4} \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x} \, dx+45 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x^2} \, dx\\ &=\frac {10}{x}+20 x+20 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x^3} \, dx-\frac {155}{4} \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x} \, dx+45 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{x^2} \, dx-150 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} x}{\left (-4+x+x^2\right )^2} \, dx-170 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{\left (-4+x+x^2\right )^2} \, dx+580 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}} x}{\left (-4+x+x^2\right )^3} \, dx+3860 \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{\left (-4+x+x^2\right )^3} \, dx+\frac {1}{68} \left (5 \left (527-39 \sqrt {17}\right )\right ) \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{1+\sqrt {17}+2 x} \, dx+\frac {1}{68} \left (5 \left (527+39 \sqrt {17}\right )\right ) \int \frac {e^{\frac {16}{\left (-4+x+x^2\right )^2}}}{1-\sqrt {17}+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.88, size = 33, normalized size = 1.14 \begin {gather*} 10 \left (e^{\frac {16}{\left (-4+x+x^2\right )^2}} \left (-\frac {1}{x^2}-\frac {5}{x}\right )+\frac {1}{x}+2 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(640*x - 480*x^2 - 1640*x^3 + 1190*x^4 + 810*x^5 - 490*x^6 - 190*x^7 + 60*x^8 + 20*x^9 + E^(16/(16 -
8*x - 7*x^2 + 2*x^3 + x^4))*(-1280 - 1920*x + 5360*x^2 + 4540*x^3 - 1330*x^4 - 390*x^5 + 170*x^6 + 50*x^7))/(
-64*x^3 + 48*x^4 + 36*x^5 - 23*x^6 - 9*x^7 + 3*x^8 + x^9),x]

[Out]

10*(E^(16/(-4 + x + x^2)^2)*(-x^(-2) - 5/x) + x^(-1) + 2*x)

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fricas [A]  time = 0.55, size = 42, normalized size = 1.45 \begin {gather*} \frac {10 \, {\left (2 \, x^{3} - {\left (5 \, x + 1\right )} e^{\left (\frac {16}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16}\right )} + x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280)*exp(16/(x^4+2*x^3-7*x^2-8*x+16))+20
*x^9+60*x^8-190*x^7-490*x^6+810*x^5+1190*x^4-1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*
x^3),x, algorithm="fricas")

[Out]

10*(2*x^3 - (5*x + 1)*e^(16/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16)) + x)/x^2

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giac [B]  time = 0.16, size = 101, normalized size = 3.48 \begin {gather*} \frac {10 \, {\left (2 \, x^{3} - 5 \, x e^{\left (-\frac {x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16} + 1\right )} + x - e^{\left (-\frac {x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16} + 1\right )}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280)*exp(16/(x^4+2*x^3-7*x^2-8*x+16))+20
*x^9+60*x^8-190*x^7-490*x^6+810*x^5+1190*x^4-1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*
x^3),x, algorithm="giac")

[Out]

10*(2*x^3 - 5*x*e^(-(x^4 + 2*x^3 - 7*x^2 - 8*x)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) + 1) + x - e^(-(x^4 + 2*x^3 -
7*x^2 - 8*x)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) + 1))/x^2

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maple [A]  time = 0.29, size = 31, normalized size = 1.07

method	result	size
risch	\(20 x +\frac {10}{x}-\frac {10 \left (1+5 x \right ) {\mathrm e}^{\frac {16}{\left (x^{2}+x -4\right )^{2}}}}{x^{2}}\)	\(31\)
norman	\(\frac {-595 x^{4}-270 x^{3}+105 x^{6}+960 x^{2}+160 x +20 x^{7}-720 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x +470 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{2}+330 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{3}-110 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{4}-50 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}} x^{5}-160 \,{\mathrm e}^{\frac {16}{x^{4}+2 x^{3}-7 x^{2}-8 x +16}}}{x^{2} \left (x^{2}+x -4\right )^{2}}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280)*exp(16/(x^4+2*x^3-7*x^2-8*x+16))+20*x^9+6
0*x^8-190*x^7-490*x^6+810*x^5+1190*x^4-1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*x^3),x
,method=_RETURNVERBOSE)

[Out]

20*x+10/x-10*(1+5*x)/x^2*exp(16/(x^2+x-4)^2)

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maxima [B]  time = 0.55, size = 377, normalized size = 13.00 \begin {gather*} 20 \, x + \frac {10 \, {\left (567 \, x^{4} + 1284 \, x^{3} - 3013 \, x^{2} - 4466 \, x + 4624\right )}}{289 \, {\left (x^{5} + 2 \, x^{4} - 7 \, x^{3} - 8 \, x^{2} + 16 \, x\right )}} - \frac {10 \, {\left (4134 \, x^{3} - 1891 \, x^{2} - 17512 \, x + 17232\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} + \frac {30 \, {\left (924 \, x^{3} - 1793 \, x^{2} - 4696 \, x + 7536\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} + \frac {95 \, {\left (386 \, x^{3} + x^{2} - 1096 \, x + 656\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {30 \, {\left (29 \, x^{3} + 188 \, x^{2} + 9 \, x - 942\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {245 \, {\left (24 \, x^{3} - 253 \, x^{2} - 152 \, x + 496\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {405 \, {\left (14 \, x^{3} + 21 \, x^{2} + 104 \, x - 96\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {820 \, {\left (12 \, x^{3} + 18 \, x^{2} - 76 \, x - 41\right )}}{289 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {35 \, {\left (6 \, x^{3} + 9 \, x^{2} - 38 \, x + 124\right )}}{17 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )}} - \frac {10 \, {\left (5 \, x + 1\right )} e^{\left (\frac {16}{x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^7+170*x^6-390*x^5-1330*x^4+4540*x^3+5360*x^2-1920*x-1280)*exp(16/(x^4+2*x^3-7*x^2-8*x+16))+20
*x^9+60*x^8-190*x^7-490*x^6+810*x^5+1190*x^4-1640*x^3-480*x^2+640*x)/(x^9+3*x^8-9*x^7-23*x^6+36*x^5+48*x^4-64*
x^3),x, algorithm="maxima")

[Out]

20*x + 10/289*(567*x^4 + 1284*x^3 - 3013*x^2 - 4466*x + 4624)/(x^5 + 2*x^4 - 7*x^3 - 8*x^2 + 16*x) - 10/289*(4
134*x^3 - 1891*x^2 - 17512*x + 17232)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) + 30/289*(924*x^3 - 1793*x^2 - 4696*x +
7536)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) + 95/289*(386*x^3 + x^2 - 1096*x + 656)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 1
6) - 30/289*(29*x^3 + 188*x^2 + 9*x - 942)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 245/289*(24*x^3 - 253*x^2 - 152*
x + 496)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 405/289*(14*x^3 + 21*x^2 + 104*x - 96)/(x^4 + 2*x^3 - 7*x^2 - 8*x
+ 16) - 820/289*(12*x^3 + 18*x^2 - 76*x - 41)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 35/17*(6*x^3 + 9*x^2 - 38*x +
124)/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16) - 10*(5*x + 1)*e^(16/(x^4 + 2*x^3 - 7*x^2 - 8*x + 16))/x^2

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mupad [B]  time = 4.42, size = 42, normalized size = 1.45 \begin {gather*} 20\,x+\frac {10}{x}-\frac {{\mathrm {e}}^{\frac {16}{x^4+2\,x^3-7\,x^2-8\,x+16}}\,\left (50\,x+10\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((640*x - exp(16/(2*x^3 - 7*x^2 - 8*x + x^4 + 16))*(1920*x - 5360*x^2 - 4540*x^3 + 1330*x^4 + 390*x^5 - 170
*x^6 - 50*x^7 + 1280) - 480*x^2 - 1640*x^3 + 1190*x^4 + 810*x^5 - 490*x^6 - 190*x^7 + 60*x^8 + 20*x^9)/(48*x^4
- 64*x^3 + 36*x^5 - 23*x^6 - 9*x^7 + 3*x^8 + x^9),x)

[Out]

20*x + 10/x - (exp(16/(2*x^3 - 7*x^2 - 8*x + x^4 + 16))*(50*x + 10))/x^2

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sympy [A]  time = 0.24, size = 37, normalized size = 1.28 \begin {gather*} 20 x + \frac {10}{x} + \frac {\left (- 50 x - 10\right ) e^{\frac {16}{x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 16}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x**7+170*x**6-390*x**5-1330*x**4+4540*x**3+5360*x**2-1920*x-1280)*exp(16/(x**4+2*x**3-7*x**2-8*
x+16))+20*x**9+60*x**8-190*x**7-490*x**6+810*x**5+1190*x**4-1640*x**3-480*x**2+640*x)/(x**9+3*x**8-9*x**7-23*x
**6+36*x**5+48*x**4-64*x**3),x)

[Out]

20*x + 10/x + (-50*x - 10)*exp(16/(x**4 + 2*x**3 - 7*x**2 - 8*x + 16))/x**2

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