3.97.67 \(\int \frac {e^{\frac {16 x-81 x^2-24 e^{\frac {1}{5} (5 x+\log (5))} x^2+9 e^{\frac {2}{5} (5 x+\log (5))} x^3}{16-24 e^{\frac {1}{5} (5 x+\log (5))} x+9 e^{\frac {2}{5} (5 x+\log (5))} x^2}} (-64+648 x-108 e^{\frac {2}{5} (5 x+\log (5))} x^2+27 e^{\frac {3}{5} (5 x+\log (5))} x^3+e^{\frac {1}{5} (5 x+\log (5))} (144 x+486 x^3))}{-64+144 e^{\frac {1}{5} (5 x+\log (5))} x-108 e^{\frac {2}{5} (5 x+\log (5))} x^2+27 e^{\frac {3}{5} (5 x+\log (5))} x^3} \, dx\)

Optimal. Leaf size=26 \[ e^{-\frac {9}{\left (-\sqrt [5]{5} e^x+\frac {4}{3 x}\right )^2}+x} \]

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Rubi [F]  time = 39.68, 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 {16 x-81 x^2-24 e^{\frac {1}{5} (5 x+\log (5))} x^2+9 e^{\frac {2}{5} (5 x+\log (5))} x^3}{16-24 e^{\frac {1}{5} (5 x+\log (5))} x+9 e^{\frac {2}{5} (5 x+\log (5))} x^2}\right ) \left (-64+648 x-108 e^{\frac {2}{5} (5 x+\log (5))} x^2+27 e^{\frac {3}{5} (5 x+\log (5))} x^3+e^{\frac {1}{5} (5 x+\log (5))} \left (144 x+486 x^3\right )\right )}{-64+144 e^{\frac {1}{5} (5 x+\log (5))} x-108 e^{\frac {2}{5} (5 x+\log (5))} x^2+27 e^{\frac {3}{5} (5 x+\log (5))} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((16*x - 81*x^2 - 24*E^((5*x + Log[5])/5)*x^2 + 9*E^((2*(5*x + Log[5]))/5)*x^3)/(16 - 24*E^((5*x + Log[
5])/5)*x + 9*E^((2*(5*x + Log[5]))/5)*x^2))*(-64 + 648*x - 108*E^((2*(5*x + Log[5]))/5)*x^2 + 27*E^((3*(5*x +
Log[5]))/5)*x^3 + E^((5*x + Log[5])/5)*(144*x + 486*x^3)))/(-64 + 144*E^((5*x + Log[5])/5)*x - 108*E^((2*(5*x
+ Log[5]))/5)*x^2 + 27*E^((3*(5*x + Log[5]))/5)*x^3),x]

[Out]

Defer[Int][E^((x*(16 - 3*(27 + 8*5^(1/5)*E^x)*x + 9*5^(2/5)*E^(2*x)*x^2))/(4 - 3*5^(1/5)*E^x*x)^2), x] + 648*D
efer[Int][(E^((x*(16 - 3*(27 + 8*5^(1/5)*E^x)*x + 9*5^(2/5)*E^(2*x)*x^2))/(4 - 3*5^(1/5)*E^x*x)^2)*x)/(-4 + 3*
5^(1/5)*E^x*x)^3, x] + 648*Defer[Int][(E^((x*(16 - 3*(27 + 8*5^(1/5)*E^x)*x + 9*5^(2/5)*E^(2*x)*x^2))/(4 - 3*5
^(1/5)*E^x*x)^2)*x^2)/(-4 + 3*5^(1/5)*E^x*x)^3, x] + 162*Defer[Int][(E^((x*(16 - 3*(27 + 8*5^(1/5)*E^x)*x + 9*
5^(2/5)*E^(2*x)*x^2))/(4 - 3*5^(1/5)*E^x*x)^2)*x^2)/(-4 + 3*5^(1/5)*E^x*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) \left (64-72 \left (9+2 \sqrt [5]{5} e^x\right ) x+108\ 5^{2/5} e^{2 x} x^2-27 \sqrt [5]{5} e^x \left (18+5^{2/5} e^{2 x}\right ) x^3\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^3} \, dx\\ &=\int \left (\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right )+\frac {648 \exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x (1+x)}{\left (-4+3 \sqrt [5]{5} e^x x\right )^3}+\frac {162 \exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x^2}{\left (-4+3 \sqrt [5]{5} e^x x\right )^2}\right ) \, dx\\ &=162 \int \frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x^2}{\left (-4+3 \sqrt [5]{5} e^x x\right )^2} \, dx+648 \int \frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x (1+x)}{\left (-4+3 \sqrt [5]{5} e^x x\right )^3} \, dx+\int \exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) \, dx\\ &=162 \int \frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x^2}{\left (-4+3 \sqrt [5]{5} e^x x\right )^2} \, dx+648 \int \left (\frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x}{\left (-4+3 \sqrt [5]{5} e^x x\right )^3}+\frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x^2}{\left (-4+3 \sqrt [5]{5} e^x x\right )^3}\right ) \, dx+\int \exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) \, dx\\ &=162 \int \frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x^2}{\left (-4+3 \sqrt [5]{5} e^x x\right )^2} \, dx+648 \int \frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x}{\left (-4+3 \sqrt [5]{5} e^x x\right )^3} \, dx+648 \int \frac {\exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) x^2}{\left (-4+3 \sqrt [5]{5} e^x x\right )^3} \, dx+\int \exp \left (\frac {x \left (16-3 \left (27+8 \sqrt [5]{5} e^x\right ) x+9\ 5^{2/5} e^{2 x} x^2\right )}{\left (4-3 \sqrt [5]{5} e^x x\right )^2}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 24, normalized size = 0.92 \begin {gather*} e^{x-\frac {81 x^2}{\left (-4+3 \sqrt [5]{5} e^x x\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((16*x - 81*x^2 - 24*E^((5*x + Log[5])/5)*x^2 + 9*E^((2*(5*x + Log[5]))/5)*x^3)/(16 - 24*E^((5*x
+ Log[5])/5)*x + 9*E^((2*(5*x + Log[5]))/5)*x^2))*(-64 + 648*x - 108*E^((2*(5*x + Log[5]))/5)*x^2 + 27*E^((3*(
5*x + Log[5]))/5)*x^3 + E^((5*x + Log[5])/5)*(144*x + 486*x^3)))/(-64 + 144*E^((5*x + Log[5])/5)*x - 108*E^((2
*(5*x + Log[5]))/5)*x^2 + 27*E^((3*(5*x + Log[5]))/5)*x^3),x]

[Out]

E^(x - (81*x^2)/(-4 + 3*5^(1/5)*E^x*x)^2)

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fricas [B]  time = 0.49, size = 65, normalized size = 2.50 \begin {gather*} e^{\left (\frac {9 \, x^{3} e^{\left (2 \, x + \frac {2}{5} \, \log \relax (5)\right )} - 24 \, x^{2} e^{\left (x + \frac {1}{5} \, \log \relax (5)\right )} - 81 \, x^{2} + 16 \, x}{9 \, x^{2} e^{\left (2 \, x + \frac {2}{5} \, \log \relax (5)\right )} - 24 \, x e^{\left (x + \frac {1}{5} \, \log \relax (5)\right )} + 16}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^3*exp(1/5*log(5)+x)^3-108*x^2*exp(1/5*log(5)+x)^2+(486*x^3+144*x)*exp(1/5*log(5)+x)+648*x-64)*
exp((9*x^3*exp(1/5*log(5)+x)^2-24*x^2*exp(1/5*log(5)+x)-81*x^2+16*x)/(9*x^2*exp(1/5*log(5)+x)^2-24*x*exp(1/5*l
og(5)+x)+16))/(27*x^3*exp(1/5*log(5)+x)^3-108*x^2*exp(1/5*log(5)+x)^2+144*x*exp(1/5*log(5)+x)-64),x, algorithm
="fricas")

[Out]

e^((9*x^3*e^(2*x + 2/5*log(5)) - 24*x^2*e^(x + 1/5*log(5)) - 81*x^2 + 16*x)/(9*x^2*e^(2*x + 2/5*log(5)) - 24*x
*e^(x + 1/5*log(5)) + 16))

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giac [B]  time = 5.74, size = 148, normalized size = 5.69 \begin {gather*} e^{\left (\frac {9 \, x^{3} e^{\left (2 \, x + \frac {2}{5} \, \log \relax (5)\right )}}{9 \, x^{2} e^{\left (2 \, x + \frac {2}{5} \, \log \relax (5)\right )} - 24 \, x e^{\left (x + \frac {1}{5} \, \log \relax (5)\right )} + 16} - \frac {24 \, x^{2} e^{\left (x + \frac {1}{5} \, \log \relax (5)\right )}}{9 \, x^{2} e^{\left (2 \, x + \frac {2}{5} \, \log \relax (5)\right )} - 24 \, x e^{\left (x + \frac {1}{5} \, \log \relax (5)\right )} + 16} - \frac {81 \, x^{2}}{9 \, x^{2} e^{\left (2 \, x + \frac {2}{5} \, \log \relax (5)\right )} - 24 \, x e^{\left (x + \frac {1}{5} \, \log \relax (5)\right )} + 16} + \frac {16 \, x}{9 \, x^{2} e^{\left (2 \, x + \frac {2}{5} \, \log \relax (5)\right )} - 24 \, x e^{\left (x + \frac {1}{5} \, \log \relax (5)\right )} + 16}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^3*exp(1/5*log(5)+x)^3-108*x^2*exp(1/5*log(5)+x)^2+(486*x^3+144*x)*exp(1/5*log(5)+x)+648*x-64)*
exp((9*x^3*exp(1/5*log(5)+x)^2-24*x^2*exp(1/5*log(5)+x)-81*x^2+16*x)/(9*x^2*exp(1/5*log(5)+x)^2-24*x*exp(1/5*l
og(5)+x)+16))/(27*x^3*exp(1/5*log(5)+x)^3-108*x^2*exp(1/5*log(5)+x)^2+144*x*exp(1/5*log(5)+x)-64),x, algorithm
="giac")

[Out]

e^(9*x^3*e^(2*x + 2/5*log(5))/(9*x^2*e^(2*x + 2/5*log(5)) - 24*x*e^(x + 1/5*log(5)) + 16) - 24*x^2*e^(x + 1/5*
log(5))/(9*x^2*e^(2*x + 2/5*log(5)) - 24*x*e^(x + 1/5*log(5)) + 16) - 81*x^2/(9*x^2*e^(2*x + 2/5*log(5)) - 24*
x*e^(x + 1/5*log(5)) + 16) + 16*x/(9*x^2*e^(2*x + 2/5*log(5)) - 24*x*e^(x + 1/5*log(5)) + 16))

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (27 x^{3} {\mathrm e}^{\frac {3 \ln \relax (5)}{5}+3 x}-108 x^{2} {\mathrm e}^{\frac {2 \ln \relax (5)}{5}+2 x}+\left (486 x^{3}+144 x \right ) {\mathrm e}^{\frac {\ln \relax (5)}{5}+x}+648 x -64\right ) {\mathrm e}^{\frac {9 x^{3} {\mathrm e}^{\frac {2 \ln \relax (5)}{5}+2 x}-24 x^{2} {\mathrm e}^{\frac {\ln \relax (5)}{5}+x}-81 x^{2}+16 x}{9 x^{2} {\mathrm e}^{\frac {2 \ln \relax (5)}{5}+2 x}-24 x \,{\mathrm e}^{\frac {\ln \relax (5)}{5}+x}+16}}}{27 x^{3} {\mathrm e}^{\frac {3 \ln \relax (5)}{5}+3 x}-108 x^{2} {\mathrm e}^{\frac {2 \ln \relax (5)}{5}+2 x}+144 x \,{\mathrm e}^{\frac {\ln \relax (5)}{5}+x}-64}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*x^3*exp(1/5*ln(5)+x)^3-108*x^2*exp(1/5*ln(5)+x)^2+(486*x^3+144*x)*exp(1/5*ln(5)+x)+648*x-64)*exp((9*x^
3*exp(1/5*ln(5)+x)^2-24*x^2*exp(1/5*ln(5)+x)-81*x^2+16*x)/(9*x^2*exp(1/5*ln(5)+x)^2-24*x*exp(1/5*ln(5)+x)+16))
/(27*x^3*exp(1/5*ln(5)+x)^3-108*x^2*exp(1/5*ln(5)+x)^2+144*x*exp(1/5*ln(5)+x)-64),x)

[Out]

int((27*x^3*exp(1/5*ln(5)+x)^3-108*x^2*exp(1/5*ln(5)+x)^2+(486*x^3+144*x)*exp(1/5*ln(5)+x)+648*x-64)*exp((9*x^
3*exp(1/5*ln(5)+x)^2-24*x^2*exp(1/5*ln(5)+x)-81*x^2+16*x)/(9*x^2*exp(1/5*ln(5)+x)^2-24*x*exp(1/5*ln(5)+x)+16))
/(27*x^3*exp(1/5*ln(5)+x)^3-108*x^2*exp(1/5*ln(5)+x)^2+144*x*exp(1/5*ln(5)+x)-64),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^3*exp(1/5*log(5)+x)^3-108*x^2*exp(1/5*log(5)+x)^2+(486*x^3+144*x)*exp(1/5*log(5)+x)+648*x-64)*
exp((9*x^3*exp(1/5*log(5)+x)^2-24*x^2*exp(1/5*log(5)+x)-81*x^2+16*x)/(9*x^2*exp(1/5*log(5)+x)^2-24*x*exp(1/5*l
og(5)+x)+16))/(27*x^3*exp(1/5*log(5)+x)^3-108*x^2*exp(1/5*log(5)+x)^2+144*x*exp(1/5*log(5)+x)-64),x, algorithm
="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 6.62, size = 131, normalized size = 5.04 \begin {gather*} {\mathrm {e}}^{-\frac {81\,x^2}{9\,5^{2/5}\,x^2\,{\mathrm {e}}^{2\,x}-24\,5^{1/5}\,x\,{\mathrm {e}}^x+16}}\,{\mathrm {e}}^{-\frac {24\,5^{1/5}\,x^2\,{\mathrm {e}}^x}{9\,5^{2/5}\,x^2\,{\mathrm {e}}^{2\,x}-24\,5^{1/5}\,x\,{\mathrm {e}}^x+16}}\,{\mathrm {e}}^{\frac {9\,5^{2/5}\,x^3\,{\mathrm {e}}^{2\,x}}{9\,5^{2/5}\,x^2\,{\mathrm {e}}^{2\,x}-24\,5^{1/5}\,x\,{\mathrm {e}}^x+16}}\,{\mathrm {e}}^{\frac {16\,x}{9\,5^{2/5}\,x^2\,{\mathrm {e}}^{2\,x}-24\,5^{1/5}\,x\,{\mathrm {e}}^x+16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((16*x - 24*x^2*exp(x + log(5)/5) + 9*x^3*exp(2*x + (2*log(5))/5) - 81*x^2)/(9*x^2*exp(2*x + (2*log(5
))/5) - 24*x*exp(x + log(5)/5) + 16))*(648*x - 108*x^2*exp(2*x + (2*log(5))/5) + 27*x^3*exp(3*x + (3*log(5))/5
) + exp(x + log(5)/5)*(144*x + 486*x^3) - 64))/(108*x^2*exp(2*x + (2*log(5))/5) - 27*x^3*exp(3*x + (3*log(5))/
5) - 144*x*exp(x + log(5)/5) + 64),x)

[Out]

exp(-(81*x^2)/(9*5^(2/5)*x^2*exp(2*x) - 24*5^(1/5)*x*exp(x) + 16))*exp(-(24*5^(1/5)*x^2*exp(x))/(9*5^(2/5)*x^2
*exp(2*x) - 24*5^(1/5)*x*exp(x) + 16))*exp((9*5^(2/5)*x^3*exp(2*x))/(9*5^(2/5)*x^2*exp(2*x) - 24*5^(1/5)*x*exp
(x) + 16))*exp((16*x)/(9*5^(2/5)*x^2*exp(2*x) - 24*5^(1/5)*x*exp(x) + 16))

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sympy [B]  time = 3.07, size = 66, normalized size = 2.54 \begin {gather*} e^{\frac {9 \cdot 5^{\frac {2}{5}} x^{3} e^{2 x} - 24 \sqrt [5]{5} x^{2} e^{x} - 81 x^{2} + 16 x}{9 \cdot 5^{\frac {2}{5}} x^{2} e^{2 x} - 24 \sqrt [5]{5} x e^{x} + 16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x**3*exp(1/5*ln(5)+x)**3-108*x**2*exp(1/5*ln(5)+x)**2+(486*x**3+144*x)*exp(1/5*ln(5)+x)+648*x-64
)*exp((9*x**3*exp(1/5*ln(5)+x)**2-24*x**2*exp(1/5*ln(5)+x)-81*x**2+16*x)/(9*x**2*exp(1/5*ln(5)+x)**2-24*x*exp(
1/5*ln(5)+x)+16))/(27*x**3*exp(1/5*ln(5)+x)**3-108*x**2*exp(1/5*ln(5)+x)**2+144*x*exp(1/5*ln(5)+x)-64),x)

[Out]

exp((9*5**(2/5)*x**3*exp(2*x) - 24*5**(1/5)*x**2*exp(x) - 81*x**2 + 16*x)/(9*5**(2/5)*x**2*exp(2*x) - 24*5**(1
/5)*x*exp(x) + 16))

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