3.57.58 \(\int \frac {e^{\frac {216 e^3 x^2+e^2 (-864 x-432 x^2)+e (864+864 x+216 x^2)}{x^2+2 x^3+x^4+e^2 x^4+e (-2 x^3-2 x^4)}} (-432 e^4 x^3+e^2 (-4320 x-5184 x^2-1296 x^3)+e (1728+4320 x+2592 x^2+432 x^3)+e^3 (2592 x^2+1296 x^3))}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 (-3 x^5-3 x^6)+e (3 x^4+6 x^5+3 x^6)} \, dx\)

Optimal. Leaf size=21 \[ e^{\frac {216 e}{\left (x+\frac {x}{-2-x+e x}\right )^2}} \]

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Rubi [F]  time = 9.13, 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 {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}\right ) \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5-x^6+e^3 x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((216*E^3*x^2 + E^2*(-864*x - 432*x^2) + E*(864 + 864*x + 216*x^2))/(x^2 + 2*x^3 + x^4 + E^2*x^4 + E*(-
2*x^3 - 2*x^4)))*(-432*E^4*x^3 + E^2*(-4320*x - 5184*x^2 - 1296*x^3) + E*(1728 + 4320*x + 2592*x^2 + 432*x^3)
+ E^3*(2592*x^2 + 1296*x^3)))/(-x^3 - 3*x^4 - 3*x^5 - x^6 + E^3*x^6 + E^2*(-3*x^5 - 3*x^6) + E*(3*x^4 + 6*x^5
+ 3*x^6)),x]

[Out]

-1728*Defer[Int][E^((864*E + 864*(1 - E)*E*x + (1 + 216*E - 432*E^2 + 216*E^3)*x^2 + 2*(1 - E)*x^3 + (1 - E)^2
*x^4)/(x^2*(-1 + (-1 + E)*x)^2))/x^3, x] + 864*(1 - E)*Defer[Int][E^((864*E + 864*(1 - E)*E*x + (1 + 216*E - 4
32*E^2 + 216*E^3)*x^2 + 2*(1 - E)*x^3 + (1 - E)^2*x^4)/(x^2*(-1 + (-1 + E)*x)^2))/x^2, x] - 432*(1 - E)^3*Defe
r[Int][E^((864*E + 864*(1 - E)*E*x + (1 + 216*E - 432*E^2 + 216*E^3)*x^2 + 2*(1 - E)*x^3 + (1 - E)^2*x^4)/(x^2
*(-1 + (-1 + E)*x)^2))/(1 + (1 - E)*x)^3, x] - 864*(1 - E)^3*Defer[Int][E^((864*E + 864*(1 - E)*E*x + (1 + 216
*E - 432*E^2 + 216*E^3)*x^2 + 2*(1 - E)*x^3 + (1 - E)^2*x^4)/(x^2*(-1 + (-1 + E)*x)^2))/(1 + (1 - E)*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {216 e^3 x^2+e^2 \left (-864 x-432 x^2\right )+e \left (864+864 x+216 x^2\right )}{x^2+2 x^3+x^4+e^2 x^4+e \left (-2 x^3-2 x^4\right )}\right ) \left (-432 e^4 x^3+e^2 \left (-4320 x-5184 x^2-1296 x^3\right )+e \left (1728+4320 x+2592 x^2+432 x^3\right )+e^3 \left (2592 x^2+1296 x^3\right )\right )}{-x^3-3 x^4-3 x^5+\left (-1+e^3\right ) x^6+e^2 \left (-3 x^5-3 x^6\right )+e \left (3 x^4+6 x^5+3 x^6\right )} \, dx\\ &=\int \frac {432 \exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right ) \left (-4-10 (1-e) x-6 (1-e)^2 x^2-(1-e)^3 x^3\right )}{x^3 (1-(-1+e) x)^3} \, dx\\ &=432 \int \frac {\exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right ) \left (-4-10 (1-e) x-6 (1-e)^2 x^2-(1-e)^3 x^3\right )}{x^3 (1-(-1+e) x)^3} \, dx\\ &=432 \int \left (-\frac {4 \exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{x^3}-\frac {2 (-1+e) \exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{x^2}+\frac {(-1+e)^3 \exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{(1+(1-e) x)^3}+\frac {2 (-1+e)^3 \exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{(1+(1-e) x)^2}\right ) \, dx\\ &=-\left (1728 \int \frac {\exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{x^3} \, dx\right )+(864 (1-e)) \int \frac {\exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{x^2} \, dx-\left (432 (1-e)^3\right ) \int \frac {\exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{(1+(1-e) x)^3} \, dx-\left (864 (1-e)^3\right ) \int \frac {\exp \left (\frac {864 e+864 (1-e) e x+\left (1+216 e-432 e^2+216 e^3\right ) x^2+2 (1-e) x^3+(1-e)^2 x^4}{x^2 (-1+(-1+e) x)^2}\right )}{(1+(1-e) x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 26, normalized size = 1.24 \begin {gather*} e^{\frac {216 e (-2+(-1+e) x)^2}{x^2 (-1+(-1+e) x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((216*E^3*x^2 + E^2*(-864*x - 432*x^2) + E*(864 + 864*x + 216*x^2))/(x^2 + 2*x^3 + x^4 + E^2*x^4
+ E*(-2*x^3 - 2*x^4)))*(-432*E^4*x^3 + E^2*(-4320*x - 5184*x^2 - 1296*x^3) + E*(1728 + 4320*x + 2592*x^2 + 432
*x^3) + E^3*(2592*x^2 + 1296*x^3)))/(-x^3 - 3*x^4 - 3*x^5 - x^6 + E^3*x^6 + E^2*(-3*x^5 - 3*x^6) + E*(3*x^4 +
6*x^5 + 3*x^6)),x]

[Out]

E^((216*E*(-2 + (-1 + E)*x)^2)/(x^2*(-1 + (-1 + E)*x)^2))

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fricas [B]  time = 0.64, size = 63, normalized size = 3.00 \begin {gather*} e^{\left (\frac {216 \, {\left (x^{2} e^{3} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{2} + {\left (x^{2} + 4 \, x + 4\right )} e\right )}}{x^{4} e^{2} + x^{4} + 2 \, x^{3} + x^{2} - 2 \, {\left (x^{4} + x^{3}\right )} e}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184*x^2-4320*x)*exp(1)^2+(432*x^3+2592*x
^2+4320*x+1728)*exp(1))*exp((216*x^2*exp(1)^3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1
)^2+(-2*x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1)^2+(3*x^6+6*x^5+3*x^4)*exp(1)-x^6
-3*x^5-3*x^4-x^3),x, algorithm="fricas")

[Out]

e^(216*(x^2*e^3 - 2*(x^2 + 2*x)*e^2 + (x^2 + 4*x + 4)*e)/(x^4*e^2 + x^4 + 2*x^3 + x^2 - 2*(x^4 + x^3)*e))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184*x^2-4320*x)*exp(1)^2+(432*x^3+2592*x
^2+4320*x+1728)*exp(1))*exp((216*x^2*exp(1)^3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1
)^2+(-2*x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1)^2+(3*x^6+6*x^5+3*x^4)*exp(1)-x^6
-3*x^5-3*x^4-x^3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.63, size = 67, normalized size = 3.19

method	result	size
gosper	\({\mathrm e}^{\frac {216 \,{\mathrm e} \left (x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}-4 x \,{\mathrm e}+x^{2}+4 x +4\right )}{x^{2} \left (x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}-2 x \,{\mathrm e}+x^{2}+2 x +1\right )}}\)	\(67\)
risch	\({\mathrm e}^{-\frac {216 \left (-2 x^{2} {\mathrm e}^{2}+x^{2} {\mathrm e}+x^{2} {\mathrm e}^{3}-4 \,{\mathrm e}^{2} x +4 x \,{\mathrm e}+4 \,{\mathrm e}\right )}{x^{2} \left (2 x^{2} {\mathrm e}-x^{2} {\mathrm e}^{2}+2 x \,{\mathrm e}-x^{2}-2 x -1\right )}}\)	\(72\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {216 x^{2} {\mathrm e}^{3}+\left (-432 x^{2}-864 x \right ) {\mathrm e}^{2}+\left (216 x^{2}+864 x +864\right ) {\mathrm e}}{x^{4} {\mathrm e}^{2}+\left (-2 x^{4}-2 x^{3}\right ) {\mathrm e}+x^{4}+2 x^{3}+x^{2}}}+\left (-2 \,{\mathrm e}+2\right ) x^{3} {\mathrm e}^{\frac {216 x^{2} {\mathrm e}^{3}+\left (-432 x^{2}-864 x \right ) {\mathrm e}^{2}+\left (216 x^{2}+864 x +864\right ) {\mathrm e}}{x^{4} {\mathrm e}^{2}+\left (-2 x^{4}-2 x^{3}\right ) {\mathrm e}+x^{4}+2 x^{3}+x^{2}}}+\left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) x^{4} {\mathrm e}^{\frac {216 x^{2} {\mathrm e}^{3}+\left (-432 x^{2}-864 x \right ) {\mathrm e}^{2}+\left (216 x^{2}+864 x +864\right ) {\mathrm e}}{x^{4} {\mathrm e}^{2}+\left (-2 x^{4}-2 x^{3}\right ) {\mathrm e}+x^{4}+2 x^{3}+x^{2}}}}{x^{2} \left (x \,{\mathrm e}-x -1\right )^{2}}\)	\(270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184*x^2-4320*x)*exp(1)^2+(432*x^3+2592*x^2+432
0*x+1728)*exp(1))*exp((216*x^2*exp(1)^3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1)^2+(-
2*x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1)^2+(3*x^6+6*x^5+3*x^4)*exp(1)-x^6-3*x^5
-3*x^4-x^3),x,method=_RETURNVERBOSE)

[Out]

exp(216*exp(1)*(x^2*exp(1)^2-2*x^2*exp(1)-4*x*exp(1)+x^2+4*x+4)/x^2/(x^2*exp(1)^2-2*x^2*exp(1)-2*x*exp(1)+x^2+
2*x+1))

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maxima [B]  time = 2.80, size = 146, normalized size = 6.95 \begin {gather*} e^{\left (\frac {216 \, e^{3}}{x^{2} {\left (e^{2} - 2 \, e + 1\right )} - 2 \, x {\left (e - 1\right )} + 1} - \frac {864 \, e^{3}}{x {\left (e - 1\right )} - 1} - \frac {432 \, e^{2}}{x^{2} {\left (e^{2} - 2 \, e + 1\right )} - 2 \, x {\left (e - 1\right )} + 1} + \frac {1728 \, e^{2}}{x {\left (e - 1\right )} - 1} + \frac {864 \, e^{2}}{x} + \frac {216 \, e}{x^{2} {\left (e^{2} - 2 \, e + 1\right )} - 2 \, x {\left (e - 1\right )} + 1} - \frac {864 \, e}{x {\left (e - 1\right )} - 1} - \frac {864 \, e}{x} + \frac {864 \, e}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*x^3*exp(1)^4+(1296*x^3+2592*x^2)*exp(1)^3+(-1296*x^3-5184*x^2-4320*x)*exp(1)^2+(432*x^3+2592*x
^2+4320*x+1728)*exp(1))*exp((216*x^2*exp(1)^3+(-432*x^2-864*x)*exp(1)^2+(216*x^2+864*x+864)*exp(1))/(x^4*exp(1
)^2+(-2*x^4-2*x^3)*exp(1)+x^4+2*x^3+x^2))/(x^6*exp(1)^3+(-3*x^6-3*x^5)*exp(1)^2+(3*x^6+6*x^5+3*x^4)*exp(1)-x^6
-3*x^5-3*x^4-x^3),x, algorithm="maxima")

[Out]

e^(216*e^3/(x^2*(e^2 - 2*e + 1) - 2*x*(e - 1) + 1) - 864*e^3/(x*(e - 1) - 1) - 432*e^2/(x^2*(e^2 - 2*e + 1) -
2*x*(e - 1) + 1) + 1728*e^2/(x*(e - 1) - 1) + 864*e^2/x + 216*e/(x^2*(e^2 - 2*e + 1) - 2*x*(e - 1) + 1) - 864*
e/(x*(e - 1) - 1) - 864*e/x + 864*e/x^2)

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mupad [B]  time = 4.44, size = 213, normalized size = 10.14 \begin {gather*} {\mathrm {e}}^{\frac {216\,\mathrm {e}}{2\,x-2\,x\,\mathrm {e}-2\,x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2+x^2+1}}\,{\mathrm {e}}^{\frac {216\,{\mathrm {e}}^3}{2\,x-2\,x\,\mathrm {e}-2\,x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2+x^2+1}}\,{\mathrm {e}}^{-\frac {432\,{\mathrm {e}}^2}{2\,x-2\,x\,\mathrm {e}-2\,x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2+x^2+1}}\,{\mathrm {e}}^{\frac {864\,\mathrm {e}}{x-2\,x^2\,\mathrm {e}-2\,x^3\,\mathrm {e}+x^3\,{\mathrm {e}}^2+2\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {864\,{\mathrm {e}}^2}{x-2\,x^2\,\mathrm {e}-2\,x^3\,\mathrm {e}+x^3\,{\mathrm {e}}^2+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {864\,\mathrm {e}}{x^4\,{\mathrm {e}}^2-2\,x^4\,\mathrm {e}-2\,x^3\,\mathrm {e}+x^2+2\,x^3+x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(1)*(864*x + 216*x^2 + 864) - exp(2)*(864*x + 432*x^2) + 216*x^2*exp(3))/(x^4*exp(2) - exp(1)*(2
*x^3 + 2*x^4) + x^2 + 2*x^3 + x^4))*(exp(1)*(4320*x + 2592*x^2 + 432*x^3 + 1728) - exp(2)*(4320*x + 5184*x^2 +
1296*x^3) + exp(3)*(2592*x^2 + 1296*x^3) - 432*x^3*exp(4)))/(exp(2)*(3*x^5 + 3*x^6) - x^6*exp(3) - exp(1)*(3*
x^4 + 6*x^5 + 3*x^6) + x^3 + 3*x^4 + 3*x^5 + x^6),x)

[Out]

exp((216*exp(1))/(2*x - 2*x*exp(1) - 2*x^2*exp(1) + x^2*exp(2) + x^2 + 1))*exp((216*exp(3))/(2*x - 2*x*exp(1)
- 2*x^2*exp(1) + x^2*exp(2) + x^2 + 1))*exp(-(432*exp(2))/(2*x - 2*x*exp(1) - 2*x^2*exp(1) + x^2*exp(2) + x^2
+ 1))*exp((864*exp(1))/(x - 2*x^2*exp(1) - 2*x^3*exp(1) + x^3*exp(2) + 2*x^2 + x^3))*exp(-(864*exp(2))/(x - 2*
x^2*exp(1) - 2*x^3*exp(1) + x^3*exp(2) + 2*x^2 + x^3))*exp((864*exp(1))/(x^4*exp(2) - 2*x^4*exp(1) - 2*x^3*exp
(1) + x^2 + 2*x^3 + x^4))

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sympy [B]  time = 2.66, size = 70, normalized size = 3.33 \begin {gather*} e^{\frac {216 x^{2} e^{3} + \left (- 432 x^{2} - 864 x\right ) e^{2} + e \left (216 x^{2} + 864 x + 864\right )}{x^{4} + x^{4} e^{2} + 2 x^{3} + x^{2} + e \left (- 2 x^{4} - 2 x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*x**3*exp(1)**4+(1296*x**3+2592*x**2)*exp(1)**3+(-1296*x**3-5184*x**2-4320*x)*exp(1)**2+(432*x*
*3+2592*x**2+4320*x+1728)*exp(1))*exp((216*x**2*exp(1)**3+(-432*x**2-864*x)*exp(1)**2+(216*x**2+864*x+864)*exp
(1))/(x**4*exp(1)**2+(-2*x**4-2*x**3)*exp(1)+x**4+2*x**3+x**2))/(x**6*exp(1)**3+(-3*x**6-3*x**5)*exp(1)**2+(3*
x**6+6*x**5+3*x**4)*exp(1)-x**6-3*x**5-3*x**4-x**3),x)

[Out]

exp((216*x**2*exp(3) + (-432*x**2 - 864*x)*exp(2) + E*(216*x**2 + 864*x + 864))/(x**4 + x**4*exp(2) + 2*x**3 +
x**2 + E*(-2*x**4 - 2*x**3)))

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