3.94.17 \(\int \frac {e^{\frac {x^2-50 x^3+645 x^4-502 x^5+150 x^6-20 x^7+x^8+e^{2 x} (625-500 x+150 x^2-20 x^3+x^4)+e^x (-50 x+1270 x^2-1002 x^3+300 x^4-40 x^5+2 x^6)}{x^2}} (-50 x^3+1290 x^4-1506 x^5+600 x^6-100 x^7+6 x^8+e^{2 x} (-1250+1750 x-1000 x^2+280 x^3-38 x^4+2 x^5)+e^x (50 x-50 x^2+268 x^3-402 x^4+180 x^5-32 x^6+2 x^7))}{5 x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{5} e^{\left (-1+(5-x)^2 \left (\frac {e^x}{x}+x\right )\right )^2} \]

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Rubi [F]  time = 22.19, 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 {x^2-50 x^3+645 x^4-502 x^5+150 x^6-20 x^7+x^8+e^{2 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^x \left (-50 x+1270 x^2-1002 x^3+300 x^4-40 x^5+2 x^6\right )}{x^2}\right ) \left (-50 x^3+1290 x^4-1506 x^5+600 x^6-100 x^7+6 x^8+e^{2 x} \left (-1250+1750 x-1000 x^2+280 x^3-38 x^4+2 x^5\right )+e^x \left (50 x-50 x^2+268 x^3-402 x^4+180 x^5-32 x^6+2 x^7\right )\right )}{5 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((x^2 - 50*x^3 + 645*x^4 - 502*x^5 + 150*x^6 - 20*x^7 + x^8 + E^(2*x)*(625 - 500*x + 150*x^2 - 20*x^3 +
x^4) + E^x*(-50*x + 1270*x^2 - 1002*x^3 + 300*x^4 - 40*x^5 + 2*x^6))/x^2)*(-50*x^3 + 1290*x^4 - 1506*x^5 + 60
0*x^6 - 100*x^7 + 6*x^8 + E^(2*x)*(-1250 + 1750*x - 1000*x^2 + 280*x^3 - 38*x^4 + 2*x^5) + E^x*(50*x - 50*x^2
+ 268*x^3 - 402*x^4 + 180*x^5 - 32*x^6 + 2*x^7)))/(5*x^3),x]

[Out]

-10*Defer[Int][E^((E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2), x] + (268*Defer[Int][E^(x + (E^x*(-5
+ x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2), x])/5 + 56*Defer[Int][E^(2*x + (E^x*(-5 + x)^2 + x*(-1 + 25*x
- 10*x^2 + x^3))^2/x^2), x] - 250*Defer[Int][E^(2*x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)/x
^3, x] + 10*Defer[Int][E^(x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)/x^2, x] + 350*Defer[Int][
E^(2*x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)/x^2, x] - 10*Defer[Int][E^(x + (E^x*(-5 + x)^2
+ x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)/x, x] - 200*Defer[Int][E^(2*x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^
2 + x^3))^2/x^2)/x, x] + 258*Defer[Int][E^((E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)*x, x] - (402
*Defer[Int][E^(x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)*x, x])/5 - (38*Defer[Int][E^(2*x + (
E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)*x, x])/5 - (1506*Defer[Int][E^((E^x*(-5 + x)^2 + x*(-1 +
25*x - 10*x^2 + x^3))^2/x^2)*x^2, x])/5 + 36*Defer[Int][E^(x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3)
)^2/x^2)*x^2, x] + (2*Defer[Int][E^(2*x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)*x^2, x])/5 +
120*Defer[Int][E^((E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)*x^3, x] - (32*Defer[Int][E^(x + (E^x*
(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)*x^3, x])/5 - 20*Defer[Int][E^((E^x*(-5 + x)^2 + x*(-1 + 25*x
- 10*x^2 + x^3))^2/x^2)*x^4, x] + (2*Defer[Int][E^(x + (E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)
*x^4, x])/5 + (6*Defer[Int][E^((E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)*x^5, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {\exp \left (\frac {x^2-50 x^3+645 x^4-502 x^5+150 x^6-20 x^7+x^8+e^{2 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^x \left (-50 x+1270 x^2-1002 x^3+300 x^4-40 x^5+2 x^6\right )}{x^2}\right ) \left (-50 x^3+1290 x^4-1506 x^5+600 x^6-100 x^7+6 x^8+e^{2 x} \left (-1250+1750 x-1000 x^2+280 x^3-38 x^4+2 x^5\right )+e^x \left (50 x-50 x^2+268 x^3-402 x^4+180 x^5-32 x^6+2 x^7\right )\right )}{x^3} \, dx\\ &=\frac {1}{5} \int \frac {2 \exp \left (\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (5-x) \left (-e^{2 x} (-5+x)^2 \left (5-4 x+x^2\right )-x^3 \left (5-128 x+125 x^2-35 x^3+3 x^4\right )-e^x x \left (-5+4 x-26 x^2+35 x^3-11 x^4+x^5\right )\right )}{x^3} \, dx\\ &=\frac {2}{5} \int \frac {\exp \left (\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (5-x) \left (-e^{2 x} (-5+x)^2 \left (5-4 x+x^2\right )-x^3 \left (5-128 x+125 x^2-35 x^3+3 x^4\right )-e^x x \left (-5+4 x-26 x^2+35 x^3-11 x^4+x^5\right )\right )}{x^3} \, dx\\ &=\frac {2}{5} \int \left (\frac {\exp \left (2 x+\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (-5+x)^3 \left (5-4 x+x^2\right )}{x^3}+\exp \left (\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (-5+x) (-5+3 x) \left (-1+25 x-10 x^2+x^3\right )+\frac {\exp \left (x+\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (-5+x) \left (-5+4 x-26 x^2+35 x^3-11 x^4+x^5\right )}{x^2}\right ) \, dx\\ &=\frac {2}{5} \int \frac {\exp \left (2 x+\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (-5+x)^3 \left (5-4 x+x^2\right )}{x^3} \, dx+\frac {2}{5} \int \exp \left (\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (-5+x) (-5+3 x) \left (-1+25 x-10 x^2+x^3\right ) \, dx+\frac {2}{5} \int \frac {\exp \left (x+\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}\right ) (-5+x) \left (-5+4 x-26 x^2+35 x^3-11 x^4+x^5\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 = 0.19, size = 37, normalized size = 1.37 \begin {gather*} \frac {1}{5} e^{\frac {\left (e^x (-5+x)^2+x \left (-1+25 x-10 x^2+x^3\right )\right )^2}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((x^2 - 50*x^3 + 645*x^4 - 502*x^5 + 150*x^6 - 20*x^7 + x^8 + E^(2*x)*(625 - 500*x + 150*x^2 - 20
*x^3 + x^4) + E^x*(-50*x + 1270*x^2 - 1002*x^3 + 300*x^4 - 40*x^5 + 2*x^6))/x^2)*(-50*x^3 + 1290*x^4 - 1506*x^
5 + 600*x^6 - 100*x^7 + 6*x^8 + E^(2*x)*(-1250 + 1750*x - 1000*x^2 + 280*x^3 - 38*x^4 + 2*x^5) + E^x*(50*x - 5
0*x^2 + 268*x^3 - 402*x^4 + 180*x^5 - 32*x^6 + 2*x^7)))/(5*x^3),x]

[Out]

E^((E^x*(-5 + x)^2 + x*(-1 + 25*x - 10*x^2 + x^3))^2/x^2)/5

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fricas [B]  time = 0.59, size = 93, normalized size = 3.44 \begin {gather*} \frac {1}{5} \, e^{\left (\frac {x^{8} - 20 \, x^{7} + 150 \, x^{6} - 502 \, x^{5} + 645 \, x^{4} - 50 \, x^{3} + x^{2} + {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{6} - 20 \, x^{5} + 150 \, x^{4} - 501 \, x^{3} + 635 \, x^{2} - 25 \, x\right )} e^{x}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((2*x^5-38*x^4+280*x^3-1000*x^2+1750*x-1250)*exp(x)^2+(2*x^7-32*x^6+180*x^5-402*x^4+268*x^3-50*x
^2+50*x)*exp(x)+6*x^8-100*x^7+600*x^6-1506*x^5+1290*x^4-50*x^3)*exp(((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^2+(
2*x^6-40*x^5+300*x^4-1002*x^3+1270*x^2-50*x)*exp(x)+x^8-20*x^7+150*x^6-502*x^5+645*x^4-50*x^3+x^2)/x^2)/x^3,x,
algorithm="fricas")

[Out]

1/5*e^((x^8 - 20*x^7 + 150*x^6 - 502*x^5 + 645*x^4 - 50*x^3 + x^2 + (x^4 - 20*x^3 + 150*x^2 - 500*x + 625)*e^(
2*x) + 2*(x^6 - 20*x^5 + 150*x^4 - 501*x^3 + 635*x^2 - 25*x)*e^x)/x^2)

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giac [B]  time = 1.14, size = 107, normalized size = 3.96 \begin {gather*} \frac {1}{5} \, e^{\left (x^{6} - 20 \, x^{5} + 2 \, x^{4} e^{x} + 150 \, x^{4} - 40 \, x^{3} e^{x} - 502 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 300 \, x^{2} e^{x} + 645 \, x^{2} - 20 \, x e^{\left (2 \, x\right )} - 1002 \, x e^{x} - 50 \, x - \frac {500 \, e^{\left (2 \, x\right )}}{x} - \frac {50 \, e^{x}}{x} + \frac {625 \, e^{\left (2 \, x\right )}}{x^{2}} + 150 \, e^{\left (2 \, x\right )} + 1270 \, e^{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((2*x^5-38*x^4+280*x^3-1000*x^2+1750*x-1250)*exp(x)^2+(2*x^7-32*x^6+180*x^5-402*x^4+268*x^3-50*x
^2+50*x)*exp(x)+6*x^8-100*x^7+600*x^6-1506*x^5+1290*x^4-50*x^3)*exp(((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^2+(
2*x^6-40*x^5+300*x^4-1002*x^3+1270*x^2-50*x)*exp(x)+x^8-20*x^7+150*x^6-502*x^5+645*x^4-50*x^3+x^2)/x^2)/x^3,x,
algorithm="giac")

[Out]

1/5*e^(x^6 - 20*x^5 + 2*x^4*e^x + 150*x^4 - 40*x^3*e^x - 502*x^3 + x^2*e^(2*x) + 300*x^2*e^x + 645*x^2 - 20*x*
e^(2*x) - 1002*x*e^x - 50*x - 500*e^(2*x)/x - 50*e^x/x + 625*e^(2*x)/x^2 + 150*e^(2*x) + 1270*e^x + 1)

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maple [B]  time = 0.19, size = 119, normalized size = 4.41

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {x^{8}+2 x^{6} {\mathrm e}^{x}-20 x^{7}-40 x^{5} {\mathrm e}^{x}+150 x^{6}+{\mathrm e}^{2 x} x^{4}+300 \,{\mathrm e}^{x} x^{4}-502 x^{5}-20 \,{\mathrm e}^{2 x} x^{3}-1002 \,{\mathrm e}^{x} x^{3}+645 x^{4}+150 \,{\mathrm e}^{2 x} x^{2}+1270 \,{\mathrm e}^{x} x^{2}-50 x^{3}-500 x \,{\mathrm e}^{2 x}-50 \,{\mathrm e}^{x} x +x^{2}+625 \,{\mathrm e}^{2 x}}{x^{2}}}}{5}\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((2*x^5-38*x^4+280*x^3-1000*x^2+1750*x-1250)*exp(x)^2+(2*x^7-32*x^6+180*x^5-402*x^4+268*x^3-50*x^2+50*
x)*exp(x)+6*x^8-100*x^7+600*x^6-1506*x^5+1290*x^4-50*x^3)*exp(((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^2+(2*x^6-
40*x^5+300*x^4-1002*x^3+1270*x^2-50*x)*exp(x)+x^8-20*x^7+150*x^6-502*x^5+645*x^4-50*x^3+x^2)/x^2)/x^3,x,method
=_RETURNVERBOSE)

[Out]

1/5*exp((x^8+2*x^6*exp(x)-20*x^7-40*x^5*exp(x)+150*x^6+exp(2*x)*x^4+300*exp(x)*x^4-502*x^5-20*exp(2*x)*x^3-100
2*exp(x)*x^3+645*x^4+150*exp(2*x)*x^2+1270*exp(x)*x^2-50*x^3-500*x*exp(2*x)-50*exp(x)*x+x^2+625*exp(2*x))/x^2)

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maxima [B]  time = 1.19, size = 107, normalized size = 3.96 \begin {gather*} \frac {1}{5} \, e^{\left (x^{6} - 20 \, x^{5} + 2 \, x^{4} e^{x} + 150 \, x^{4} - 40 \, x^{3} e^{x} - 502 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 300 \, x^{2} e^{x} + 645 \, x^{2} - 20 \, x e^{\left (2 \, x\right )} - 1002 \, x e^{x} - 50 \, x - \frac {500 \, e^{\left (2 \, x\right )}}{x} - \frac {50 \, e^{x}}{x} + \frac {625 \, e^{\left (2 \, x\right )}}{x^{2}} + 150 \, e^{\left (2 \, x\right )} + 1270 \, e^{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((2*x^5-38*x^4+280*x^3-1000*x^2+1750*x-1250)*exp(x)^2+(2*x^7-32*x^6+180*x^5-402*x^4+268*x^3-50*x
^2+50*x)*exp(x)+6*x^8-100*x^7+600*x^6-1506*x^5+1290*x^4-50*x^3)*exp(((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^2+(
2*x^6-40*x^5+300*x^4-1002*x^3+1270*x^2-50*x)*exp(x)+x^8-20*x^7+150*x^6-502*x^5+645*x^4-50*x^3+x^2)/x^2)/x^3,x,
algorithm="maxima")

[Out]

1/5*e^(x^6 - 20*x^5 + 2*x^4*e^x + 150*x^4 - 40*x^3*e^x - 502*x^3 + x^2*e^(2*x) + 300*x^2*e^x + 645*x^2 - 20*x*
e^(2*x) - 1002*x*e^x - 50*x - 500*e^(2*x)/x - 50*e^x/x + 625*e^(2*x)/x^2 + 150*e^(2*x) + 1270*e^x + 1)

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mupad [B]  time = 9.17, size = 123, normalized size = 4.56 \begin {gather*} \frac {{\mathrm {e}}^{150\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-1002\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-50\,x}\,{\mathrm {e}}^{x^6}\,\mathrm {e}\,{\mathrm {e}}^{-20\,x\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{2\,x^4\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-40\,x^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-\frac {50\,{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{300\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-20\,x^5}\,{\mathrm {e}}^{150\,x^4}\,{\mathrm {e}}^{-502\,x^3}\,{\mathrm {e}}^{645\,x^2}\,{\mathrm {e}}^{1270\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-\frac {500\,{\mathrm {e}}^{2\,x}}{x}}\,{\mathrm {e}}^{\frac {625\,{\mathrm {e}}^{2\,x}}{x^2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(2*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(x)*(50*x - 1270*x^2 + 1002*x^3 - 300*x^4 + 40*
x^5 - 2*x^6) + x^2 - 50*x^3 + 645*x^4 - 502*x^5 + 150*x^6 - 20*x^7 + x^8)/x^2)*(exp(2*x)*(1750*x - 1000*x^2 +
280*x^3 - 38*x^4 + 2*x^5 - 1250) + exp(x)*(50*x - 50*x^2 + 268*x^3 - 402*x^4 + 180*x^5 - 32*x^6 + 2*x^7) - 50*
x^3 + 1290*x^4 - 1506*x^5 + 600*x^6 - 100*x^7 + 6*x^8))/(5*x^3),x)

[Out]

(exp(150*exp(2*x))*exp(-1002*x*exp(x))*exp(-50*x)*exp(x^6)*exp(1)*exp(-20*x*exp(2*x))*exp(2*x^4*exp(x))*exp(-4
0*x^3*exp(x))*exp(-(50*exp(x))/x)*exp(300*x^2*exp(x))*exp(-20*x^5)*exp(150*x^4)*exp(-502*x^3)*exp(645*x^2)*exp
(1270*exp(x))*exp(x^2*exp(2*x))*exp(-(500*exp(2*x))/x)*exp((625*exp(2*x))/x^2))/5

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sympy [B]  time = 0.76, size = 94, normalized size = 3.48 \begin {gather*} \frac {e^{\frac {x^{8} - 20 x^{7} + 150 x^{6} - 502 x^{5} + 645 x^{4} - 50 x^{3} + x^{2} + \left (x^{4} - 20 x^{3} + 150 x^{2} - 500 x + 625\right ) e^{2 x} + \left (2 x^{6} - 40 x^{5} + 300 x^{4} - 1002 x^{3} + 1270 x^{2} - 50 x\right ) e^{x}}{x^{2}}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((2*x**5-38*x**4+280*x**3-1000*x**2+1750*x-1250)*exp(x)**2+(2*x**7-32*x**6+180*x**5-402*x**4+268
*x**3-50*x**2+50*x)*exp(x)+6*x**8-100*x**7+600*x**6-1506*x**5+1290*x**4-50*x**3)*exp(((x**4-20*x**3+150*x**2-5
00*x+625)*exp(x)**2+(2*x**6-40*x**5+300*x**4-1002*x**3+1270*x**2-50*x)*exp(x)+x**8-20*x**7+150*x**6-502*x**5+6
45*x**4-50*x**3+x**2)/x**2)/x**3,x)

[Out]

exp((x**8 - 20*x**7 + 150*x**6 - 502*x**5 + 645*x**4 - 50*x**3 + x**2 + (x**4 - 20*x**3 + 150*x**2 - 500*x + 6
25)*exp(2*x) + (2*x**6 - 40*x**5 + 300*x**4 - 1002*x**3 + 1270*x**2 - 50*x)*exp(x))/x**2)/5

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