3.85.29 \(\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7)}{81 x^3+54 x^5+9 x^7} \, dx\)

Optimal. Leaf size=29 \[ \frac {e^{4-\frac {1}{x}-\frac {16 x^2}{9}+\frac {x}{3+x^2}}}{x} \]

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Rubi [F]  time = 3.13, 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 {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-27 + 108*x - 12*x^3 - 16*x^5)/(27*x + 9*x^3))*(81 - 81*x + 81*x^2 - 342*x^3 - 201*x^5 - 32*x^7))/(81
*x^3 + 54*x^5 + 9*x^7),x]

[Out]

(-32*Defer[Int][E^((-27 + 108*x - 12*x^3 - 16*x^5)/(x*(27 + 9*x^2))), x])/9 + Defer[Int][E^((-27 + 108*x - 12*
x^3 - 16*x^5)/(x*(27 + 9*x^2)))/(I*Sqrt[3] - x), x]/6 + Defer[Int][E^((-27 + 108*x - 12*x^3 - 16*x^5)/(x*(27 +
9*x^2)))/x^3, x] - Defer[Int][E^((-27 + 108*x - 12*x^3 - 16*x^5)/(x*(27 + 9*x^2)))/x^2, x] + Defer[Int][E^((-
27 + 108*x - 12*x^3 - 16*x^5)/(x*(27 + 9*x^2)))/x, x]/3 - Defer[Int][E^((-27 + 108*x - 12*x^3 - 16*x^5)/(x*(27
+ 9*x^2)))/(I*Sqrt[3] + x), x]/6 - 2*Defer[Int][(E^((-27 + 108*x - 12*x^3 - 16*x^5)/(x*(27 + 9*x^2)))*x)/(3 +
x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{x^3 \left (81+54 x^2+9 x^4\right )} \, dx\\ &=9 \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{x^3 \left (27+9 x^2\right )^2} \, dx\\ &=9 \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{x^3 \left (27+9 x^2\right )^2} \, dx\\ &=9 \int \left (-\frac {32}{81} e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}+\frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{9 x^3}-\frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{9 x^2}+\frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{27 x}-\frac {2 e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} x}{9 \left (3+x^2\right )^2}-\frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} x}{27 \left (3+x^2\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x} \, dx-\frac {1}{3} \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} x}{3+x^2} \, dx-2 \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} x}{\left (3+x^2\right )^2} \, dx-\frac {32}{9} \int e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} \, dx+\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x^3} \, dx-\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x} \, dx-\frac {1}{3} \int \left (-\frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{2 \left (i \sqrt {3}-x\right )}+\frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{2 \left (i \sqrt {3}+x\right )}\right ) \, dx-2 \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} x}{\left (3+x^2\right )^2} \, dx-\frac {32}{9} \int e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} \, dx+\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x^3} \, dx-\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x^2} \, dx\\ &=\frac {1}{6} \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{i \sqrt {3}-x} \, dx-\frac {1}{6} \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{i \sqrt {3}+x} \, dx+\frac {1}{3} \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x} \, dx-2 \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} x}{\left (3+x^2\right )^2} \, dx-\frac {32}{9} \int e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}} \, dx+\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x^3} \, dx-\int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{x \left (27+9 x^2\right )}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.81, size = 29, normalized size = 1.00 \begin {gather*} \frac {e^{4-\frac {1}{x}-\frac {16 x^2}{9}+\frac {x}{3+x^2}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-27 + 108*x - 12*x^3 - 16*x^5)/(27*x + 9*x^3))*(81 - 81*x + 81*x^2 - 342*x^3 - 201*x^5 - 32*x^7
))/(81*x^3 + 54*x^5 + 9*x^7),x]

[Out]

E^(4 - x^(-1) - (16*x^2)/9 + x/(3 + x^2))/x

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fricas [A]  time = 0.74, size = 31, normalized size = 1.07 \begin {gather*} \frac {e^{\left (-\frac {16 \, x^{5} + 12 \, x^{3} - 108 \, x + 27}{9 \, {\left (x^{3} + 3 \, x\right )}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108*x-27)/(9*x^3+27*x))/(9*x^7+54*x^5+8
1*x^3),x, algorithm="fricas")

[Out]

e^(-1/9*(16*x^5 + 12*x^3 - 108*x + 27)/(x^3 + 3*x))/x

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giac [A]  time = 0.27, size = 31, normalized size = 1.07 \begin {gather*} \frac {e^{\left (-\frac {16 \, x^{5} + 12 \, x^{3} - 108 \, x + 27}{9 \, {\left (x^{3} + 3 \, x\right )}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108*x-27)/(9*x^3+27*x))/(9*x^7+54*x^5+8
1*x^3),x, algorithm="giac")

[Out]

e^(-1/9*(16*x^5 + 12*x^3 - 108*x + 27)/(x^3 + 3*x))/x

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maple [A]  time = 0.14, size = 33, normalized size = 1.14

method	result	size
gosper	\(\frac {{\mathrm e}^{-\frac {16 x^{5}+12 x^{3}-108 x +27}{9 \left (x^{2}+3\right ) x}}}{x}\)	\(33\)
risch	\(\frac {{\mathrm e}^{-\frac {16 x^{5}+12 x^{3}-108 x +27}{9 \left (x^{2}+3\right ) x}}}{x}\)	\(33\)
norman	\(\frac {x^{3} {\mathrm e}^{\frac {-16 x^{5}-12 x^{3}+108 x -27}{9 x^{3}+27 x}}+3 x \,{\mathrm e}^{\frac {-16 x^{5}-12 x^{3}+108 x -27}{9 x^{3}+27 x}}}{x^{2} \left (x^{2}+3\right )}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108*x-27)/(9*x^3+27*x))/(9*x^7+54*x^5+81*x^3)
,x,method=_RETURNVERBOSE)

[Out]

1/x*exp(-1/9*(16*x^5+12*x^3-108*x+27)/(x^2+3)/x)

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maxima [A]  time = 0.47, size = 26, normalized size = 0.90 \begin {gather*} \frac {e^{\left (-\frac {16}{9} \, x^{2} + \frac {x}{x^{2} + 3} - \frac {1}{x} + 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108*x-27)/(9*x^3+27*x))/(9*x^7+54*x^5+8
1*x^3),x, algorithm="maxima")

[Out]

e^(-16/9*x^2 + x/(x^2 + 3) - 1/x + 4)/x

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mupad [B]  time = 5.47, size = 56, normalized size = 1.93 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {3}{x^3+3\,x}}\,{\mathrm {e}}^{-\frac {4\,x^2}{3\,\left (x^2+3\right )}}\,{\mathrm {e}}^{-\frac {16\,x^4}{9\,\left (x^2+3\right )}}\,{\mathrm {e}}^{\frac {12}{x^2+3}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(12*x^3 - 108*x + 16*x^5 + 27)/(27*x + 9*x^3))*(81*x - 81*x^2 + 342*x^3 + 201*x^5 + 32*x^7 - 81))/(
81*x^3 + 54*x^5 + 9*x^7),x)

[Out]

(exp(-3/(3*x + x^3))*exp(-(4*x^2)/(3*(x^2 + 3)))*exp(-(16*x^4)/(9*(x^2 + 3)))*exp(12/(x^2 + 3)))/x

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sympy [A]  time = 0.24, size = 26, normalized size = 0.90 \begin {gather*} \frac {e^{\frac {- 16 x^{5} - 12 x^{3} + 108 x - 27}{9 x^{3} + 27 x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x**7-201*x**5-342*x**3+81*x**2-81*x+81)*exp((-16*x**5-12*x**3+108*x-27)/(9*x**3+27*x))/(9*x**7+
54*x**5+81*x**3),x)

[Out]

exp((-16*x**5 - 12*x**3 + 108*x - 27)/(9*x**3 + 27*x))/x

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