3.62.26 \(\int \frac {e^{e^{\frac {1}{2} (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2)}+\frac {1}{2} (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2)} (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} (-729 x-1215 x^2-324 x^3)+e^{\frac {x}{3+2 x}} (4374 x+6561 x^2+1944 x^3))}{9+12 x+4 x^2} \, dx\)

Optimal. Leaf size=29 \[ e^{e^{\frac {1}{2} x \left (x-81 \left (-3+e^{\frac {x}{3+2 x}}\right )^2 x\right )}} \]

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Rubi [F]  time = 46.61, 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 (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + (-728*x^2 + 486*E^(x/(3 + 2*
x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)*(-6552*x - 8736*x^2 - 2912*x^3 + E^((2*x)/(3 + 2*x))*(-729*x - 1215*x
^2 - 324*x^3) + E^(x/(3 + 2*x))*(4374*x + 6561*x^2 + 1944*x^3)))/(9 + 12*x + 4*x^2),x]

[Out]

(729*Defer[Int][E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + x/(3 + 2*x) + (-7
28*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2), x])/4 - (243*Defer[Int][E^(E^((-728*x^2 + 4
86*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + (2*x)/(3 + 2*x) + (-728*x^2 + 486*E^(x/(3 + 2*x))*x^
2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2), x])/4 - 728*Defer[Int][E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((
2*x)/(3 + 2*x))*x^2)/2) + (-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)*x, x] + 486*Def
er[Int][E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + x/(3 + 2*x) + (-728*x^2 +
486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)*x, x] - 81*Defer[Int][E^(E^((-728*x^2 + 486*E^(x/(3
+ 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + (2*x)/(3 + 2*x) + (-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((
2*x)/(3 + 2*x))*x^2)/2)*x, x] + (6561*Defer[Int][E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 +
2*x))*x^2)/2) + x/(3 + 2*x) + (-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)/(3 + 2*x)^2
, x])/4 - (2187*Defer[Int][E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + (2*x)/
(3 + 2*x) + (-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)/(3 + 2*x)^2, x])/4 - (2187*De
fer[Int][E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + x/(3 + 2*x) + (-728*x^2
+ 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)/(3 + 2*x), x])/2 + (729*Defer[Int][E^(E^((-728*x^2
+ 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + (2*x)/(3 + 2*x) + (-728*x^2 + 486*E^(x/(3 + 2*x))
*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)/(3 + 2*x), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{(3+2 x)^2} \, dx\\ &=\int \left (-728 \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x-\frac {81 \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x (3+x) (3+4 x)}{(3+2 x)^2}+\frac {243 \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x \left (18+27 x+8 x^2\right )}{(3+2 x)^2}\right ) \, dx\\ &=-\left (81 \int \frac {\exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x (3+x) (3+4 x)}{(3+2 x)^2} \, dx\right )+243 \int \frac {\exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x \left (18+27 x+8 x^2\right )}{(3+2 x)^2} \, dx-728 \int \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 40, normalized size = 1.38 \begin {gather*} e^{e^{-\frac {1}{2} \left (728-486 e^{\frac {x}{3+2 x}}+81 e^{\frac {2 x}{3+2 x}}\right ) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2) + (-728*x^2 + 486*E^(x/(
3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/2)*(-6552*x - 8736*x^2 - 2912*x^3 + E^((2*x)/(3 + 2*x))*(-729*x -
1215*x^2 - 324*x^3) + E^(x/(3 + 2*x))*(4374*x + 6561*x^2 + 1944*x^3)))/(9 + 12*x + 4*x^2),x]

[Out]

E^E^(-1/2*((728 - 486*E^(x/(3 + 2*x)) + 81*E^((2*x)/(3 + 2*x)))*x^2))

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fricas [A]  time = 0.74, size = 39, normalized size = 1.34 \begin {gather*} e^{\left (e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x^3-1215*x^2-729*x)*exp(x/(2*x+3))^2+(1944*x^3+6561*x^2+4374*x)*exp(x/(2*x+3))-2912*x^3-8736*
x^2-6552*x)*exp(-81/2*x^2*exp(x/(2*x+3))^2+243*x^2*exp(x/(2*x+3))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(2*x+3))^2+
243*x^2*exp(x/(2*x+3))-364*x^2))/(4*x^2+12*x+9),x, algorithm="fricas")

[Out]

e^(e^(-81/2*x^2*e^(2*x/(2*x + 3)) + 243*x^2*e^(x/(2*x + 3)) - 364*x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2912 \, x^{3} + 8736 \, x^{2} + 81 \, {\left (4 \, x^{3} + 15 \, x^{2} + 9 \, x\right )} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} - 243 \, {\left (8 \, x^{3} + 27 \, x^{2} + 18 \, x\right )} e^{\left (\frac {x}{2 \, x + 3}\right )} + 6552 \, x\right )} e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2} + e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2}\right )}\right )}}{4 \, x^{2} + 12 \, x + 9}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x^3-1215*x^2-729*x)*exp(x/(2*x+3))^2+(1944*x^3+6561*x^2+4374*x)*exp(x/(2*x+3))-2912*x^3-8736*
x^2-6552*x)*exp(-81/2*x^2*exp(x/(2*x+3))^2+243*x^2*exp(x/(2*x+3))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(2*x+3))^2+
243*x^2*exp(x/(2*x+3))-364*x^2))/(4*x^2+12*x+9),x, algorithm="giac")

[Out]

integrate(-(2912*x^3 + 8736*x^2 + 81*(4*x^3 + 15*x^2 + 9*x)*e^(2*x/(2*x + 3)) - 243*(8*x^3 + 27*x^2 + 18*x)*e^
(x/(2*x + 3)) + 6552*x)*e^(-81/2*x^2*e^(2*x/(2*x + 3)) + 243*x^2*e^(x/(2*x + 3)) - 364*x^2 + e^(-81/2*x^2*e^(2
*x/(2*x + 3)) + 243*x^2*e^(x/(2*x + 3)) - 364*x^2))/(4*x^2 + 12*x + 9), x)

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maple [A]  time = 0.44, size = 35, normalized size = 1.21

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {x^{2} \left (-81 \,{\mathrm e}^{\frac {2 x}{2 x +3}}+486 \,{\mathrm e}^{\frac {x}{2 x +3}}-728\right )}{2}}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-324*x^3-1215*x^2-729*x)*exp(x/(2*x+3))^2+(1944*x^3+6561*x^2+4374*x)*exp(x/(2*x+3))-2912*x^3-8736*x^2-65
52*x)*exp(-81/2*x^2*exp(x/(2*x+3))^2+243*x^2*exp(x/(2*x+3))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(2*x+3))^2+243*x^
2*exp(x/(2*x+3))-364*x^2))/(4*x^2+12*x+9),x,method=_RETURNVERBOSE)

[Out]

exp(exp(1/2*x^2*(-81*exp(2*x/(2*x+3))+486*exp(x/(2*x+3))-728)))

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maxima [A]  time = 0.74, size = 42, normalized size = 1.45 \begin {gather*} e^{\left (e^{\left (243 \, x^{2} e^{\left (-\frac {3}{2 \, {\left (2 \, x + 3\right )}} + \frac {1}{2}\right )} - \frac {81}{2} \, x^{2} e^{\left (-\frac {3}{2 \, x + 3} + 1\right )} - 364 \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x^3-1215*x^2-729*x)*exp(x/(2*x+3))^2+(1944*x^3+6561*x^2+4374*x)*exp(x/(2*x+3))-2912*x^3-8736*
x^2-6552*x)*exp(-81/2*x^2*exp(x/(2*x+3))^2+243*x^2*exp(x/(2*x+3))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(2*x+3))^2+
243*x^2*exp(x/(2*x+3))-364*x^2))/(4*x^2+12*x+9),x, algorithm="maxima")

[Out]

e^(e^(243*x^2*e^(-3/2/(2*x + 3) + 1/2) - 81/2*x^2*e^(-3/(2*x + 3) + 1) - 364*x^2))

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mupad [B]  time = 4.87, size = 41, normalized size = 1.41 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{-\frac {81\,x^2\,{\mathrm {e}}^{\frac {2\,x}{2\,x+3}}}{2}}\,{\mathrm {e}}^{243\,x^2\,{\mathrm {e}}^{\frac {x}{2\,x+3}}}\,{\mathrm {e}}^{-364\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(243*x^2*exp(x/(2*x + 3)) - (81*x^2*exp((2*x)/(2*x + 3)))/2 - 364*x^2)*exp(exp(243*x^2*exp(x/(2*x + 3
)) - (81*x^2*exp((2*x)/(2*x + 3)))/2 - 364*x^2))*(6552*x + exp((2*x)/(2*x + 3))*(729*x + 1215*x^2 + 324*x^3) -
exp(x/(2*x + 3))*(4374*x + 6561*x^2 + 1944*x^3) + 8736*x^2 + 2912*x^3))/(12*x + 4*x^2 + 9),x)

[Out]

exp(exp(-(81*x^2*exp((2*x)/(2*x + 3)))/2)*exp(243*x^2*exp(x/(2*x + 3)))*exp(-364*x^2))

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sympy [A]  time = 4.62, size = 37, normalized size = 1.28 \begin {gather*} e^{e^{- \frac {81 x^{2} e^{\frac {2 x}{2 x + 3}}}{2} + 243 x^{2} e^{\frac {x}{2 x + 3}} - 364 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-324*x**3-1215*x**2-729*x)*exp(x/(2*x+3))**2+(1944*x**3+6561*x**2+4374*x)*exp(x/(2*x+3))-2912*x**3
-8736*x**2-6552*x)*exp(-81/2*x**2*exp(x/(2*x+3))**2+243*x**2*exp(x/(2*x+3))-364*x**2)*exp(exp(-81/2*x**2*exp(x
/(2*x+3))**2+243*x**2*exp(x/(2*x+3))-364*x**2))/(4*x**2+12*x+9),x)

[Out]

exp(exp(-81*x**2*exp(2*x/(2*x + 3))/2 + 243*x**2*exp(x/(2*x + 3)) - 364*x**2))

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