3.62.97 \(\int \frac {1}{81} e^{\frac {1}{81} (81 e^{-1+2 x}+e^{-1+x} (81 x^2+11250 x^5+390625 x^8))} (162 e^{-1+2 x}+e^{-1+x} (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8)) \, dx\)

Optimal. Leaf size=23 \[ e^{e^{-1+x} \left (e^x+\left (x+\frac {625 x^4}{9}\right )^2\right )} \]

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Rubi [F]  time = 6.03, 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 {1}{81} \exp \left (\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )\right ) \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((81*E^(-1 + 2*x) + E^(-1 + x)*(81*x^2 + 11250*x^5 + 390625*x^8))/81)*(162*E^(-1 + 2*x) + E^(-1 + x)*(1
62*x + 81*x^2 + 56250*x^4 + 11250*x^5 + 3125000*x^7 + 390625*x^8)))/81,x]

[Out]

2*Defer[Int][E^(-1 + 2*x + (E^(-1 + x)*(81*E^x + 81*x^2 + 11250*x^5 + 390625*x^8))/81), x] + 2*Defer[Int][E^(-
1 + x + (E^(-1 + x)*(81*E^x + 81*x^2 + 11250*x^5 + 390625*x^8))/81)*x, x] + Defer[Int][E^(-1 + x + (E^(-1 + x)
*(81*E^x + 81*x^2 + 11250*x^5 + 390625*x^8))/81)*x^2, x] + (6250*Defer[Int][E^(-1 + x + (E^(-1 + x)*(81*E^x +
81*x^2 + 11250*x^5 + 390625*x^8))/81)*x^4, x])/9 + (1250*Defer[Int][E^(-1 + x + (E^(-1 + x)*(81*E^x + 81*x^2 +
11250*x^5 + 390625*x^8))/81)*x^5, x])/9 + (3125000*Defer[Int][E^(-1 + x + (E^(-1 + x)*(81*E^x + 81*x^2 + 1125
0*x^5 + 390625*x^8))/81)*x^7, x])/81 + (390625*Defer[Int][E^(-1 + x + (E^(-1 + x)*(81*E^x + 81*x^2 + 11250*x^5
+ 390625*x^8))/81)*x^8, x])/81

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \exp \left (\frac {1}{81} \left (81 e^{-1+2 x}+e^{-1+x} \left (81 x^2+11250 x^5+390625 x^8\right )\right )\right ) \left (162 e^{-1+2 x}+e^{-1+x} \left (162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right )\right ) \, dx\\ &=\frac {1}{81} \int \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) \left (162 e^x+162 x+81 x^2+56250 x^4+11250 x^5+3125000 x^7+390625 x^8\right ) \, dx\\ &=\frac {1}{81} \int \left (162 \exp \left (-1+2 x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right )+162 \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x+81 \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^2+56250 \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^4+11250 \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^5+3125000 \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^7+390625 \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^8\right ) \, dx\\ &=2 \int \exp \left (-1+2 x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) \, dx+2 \int \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x \, dx+\frac {1250}{9} \int \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^5 \, dx+\frac {6250}{9} \int \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^4 \, dx+\frac {390625}{81} \int \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^8 \, dx+\frac {3125000}{81} \int \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^7 \, dx+\int \exp \left (-1+x+\frac {1}{81} e^{-1+x} \left (81 e^x+81 x^2+11250 x^5+390625 x^8\right )\right ) x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.98, size = 31, normalized size = 1.35 \begin {gather*} e^{e^{-1+2 x}+\frac {1}{81} e^{-1+x} x^2 \left (9+625 x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((81*E^(-1 + 2*x) + E^(-1 + x)*(81*x^2 + 11250*x^5 + 390625*x^8))/81)*(162*E^(-1 + 2*x) + E^(-1 +
x)*(162*x + 81*x^2 + 56250*x^4 + 11250*x^5 + 3125000*x^7 + 390625*x^8)))/81,x]

[Out]

E^(E^(-1 + 2*x) + (E^(-1 + x)*x^2*(9 + 625*x^3)^2)/81)

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fricas [A]  time = 0.53, size = 30, normalized size = 1.30 \begin {gather*} e^{\left (\frac {1}{81} \, {\left (390625 \, x^{8} + 11250 \, x^{5} + 81 \, x^{2}\right )} e^{\left (x - 1\right )} + e^{\left (2 \, x - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(162*exp(x-1)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+56250*x^4+81*x^2+162*x)*exp(x-1))*exp(ex
p(x-1)*exp(x)+1/81*(390625*x^8+11250*x^5+81*x^2)*exp(x-1)),x, algorithm="fricas")

[Out]

e^(1/81*(390625*x^8 + 11250*x^5 + 81*x^2)*e^(x - 1) + e^(2*x - 1))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(162*exp(x-1)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+56250*x^4+81*x^2+162*x)*exp(x-1))*exp(ex
p(x-1)*exp(x)+1/81*(390625*x^8+11250*x^5+81*x^2)*exp(x-1)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Polynomial exponent overflow. Error: Bad Argument Value

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maple [A]  time = 0.12, size = 35, normalized size = 1.52

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{2 x -1}+\frac {390625 \,{\mathrm e}^{x -1} x^{8}}{81}+\frac {1250 \,{\mathrm e}^{x -1} x^{5}}{9}+x^{2} {\mathrm e}^{x -1}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/81*(162*exp(x-1)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+56250*x^4+81*x^2+162*x)*exp(x-1))*exp(exp(x-1)
*exp(x)+1/81*(390625*x^8+11250*x^5+81*x^2)*exp(x-1)),x,method=_RETURNVERBOSE)

[Out]

exp(exp(2*x-1)+390625/81*exp(x-1)*x^8+1250/9*exp(x-1)*x^5+x^2*exp(x-1))

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maxima [A]  time = 0.65, size = 34, normalized size = 1.48 \begin {gather*} e^{\left (\frac {390625}{81} \, x^{8} e^{\left (x - 1\right )} + \frac {1250}{9} \, x^{5} e^{\left (x - 1\right )} + x^{2} e^{\left (x - 1\right )} + e^{\left (2 \, x - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(162*exp(x-1)*exp(x)+(390625*x^8+3125000*x^7+11250*x^5+56250*x^4+81*x^2+162*x)*exp(x-1))*exp(ex
p(x-1)*exp(x)+1/81*(390625*x^8+11250*x^5+81*x^2)*exp(x-1)),x, algorithm="maxima")

[Out]

e^(390625/81*x^8*e^(x - 1) + 1250/9*x^5*e^(x - 1) + x^2*e^(x - 1) + e^(2*x - 1))

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mupad [B]  time = 4.35, size = 38, normalized size = 1.65 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-1}}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {1250\,x^5\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}{9}}\,{\mathrm {e}}^{\frac {390625\,x^8\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^x}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x - 1)*exp(x) + (exp(x - 1)*(81*x^2 + 11250*x^5 + 390625*x^8))/81)*(162*exp(x - 1)*exp(x) + exp(x
- 1)*(162*x + 81*x^2 + 56250*x^4 + 11250*x^5 + 3125000*x^7 + 390625*x^8)))/81,x)

[Out]

exp(exp(2*x)*exp(-1))*exp(x^2*exp(-1)*exp(x))*exp((1250*x^5*exp(-1)*exp(x))/9)*exp((390625*x^8*exp(-1)*exp(x))
/81)

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sympy [A]  time = 0.34, size = 32, normalized size = 1.39 \begin {gather*} e^{\frac {\left (\frac {390625 x^{8}}{81} + \frac {1250 x^{5}}{9} + x^{2}\right ) e^{x}}{e} + \frac {e^{2 x}}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*(162*exp(x-1)*exp(x)+(390625*x**8+3125000*x**7+11250*x**5+56250*x**4+81*x**2+162*x)*exp(x-1))*e
xp(exp(x-1)*exp(x)+1/81*(390625*x**8+11250*x**5+81*x**2)*exp(x-1)),x)

[Out]

exp((390625*x**8/81 + 1250*x**5/9 + x**2)*exp(-1)*exp(x) + exp(-1)*exp(2*x))

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