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

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

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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 = {} $$\begin{array}{c}\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{27x+9x^3}}(81-81x+81x^2-342x^3-201x^5-32x^7)}{81x^3+54x^5+9x^7}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{27x+9x^3}}(81-81x+81x^2-342x^3-201x^5-32x^7)}{x^3(81+54x^2+9x^4)}dx\\ & =9\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{27x+9x^3}}(81-81x+81x^2-342x^3-201x^5-32x^7)}{x^3{(27+9x^2)}^2}dx\\ & =9\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}(81-81x+81x^2-342x^3-201x^5-32x^7)}{x^3{(27+9x^2)}^2}dx\\ & =9\int (-\frac{32}{81}{e}^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}+\frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{9x^3}-\frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{9x^2}+\frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{27x}-\frac{2e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}x}{9{(3+x^2)}^2}-\frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}x}{27(3+x^2)})dx\\ & =\frac{1}{3}\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x}dx-\frac{1}{3}\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}x}{3+x^2}dx-2\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}x}{{(3+x^2)}^2}dx-\frac{32}{9}\int e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}dx+\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x^3}dx-\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x^2}dx\\ & =\frac{1}{3}\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x}dx-\frac{1}{3}\int (-\frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{2(i\sqrt{3}-x)}+\frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{2(i\sqrt{3}+x)})dx-2\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}x}{{(3+x^2)}^2}dx-\frac{32}{9}\int e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}dx+\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x^3}dx-\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x^2}dx\\ & =\frac{1}{6}\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{i\sqrt{3}-x}dx-\frac{1}{6}\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{i\sqrt{3}+x}dx+\frac{1}{3}\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x}dx-2\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}x}{{(3+x^2)}^2}dx-\frac{32}{9}\int e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}dx+\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x^3}dx-\int \frac{e^{\frac{-27+108x-12x^3-16x^5}{x(27+9x^2)}}}{x^2}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{e^{(-\frac{16x^5+12x^3-108x+27}{9(x^3+3x)})}}{x}\end{array}$$

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 $$\begin{array}{c}\frac{e^{(-\frac{16x^5+12x^3-108x+27}{9(x^3+3x)})}}{x}\end{array}$$

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	$\frac{{\mathrm{e}}^{-\frac{16x^5+12x^3-108x+27}{9(x^2+3)x}}}{x}$	$33$
risch	$\frac{{\mathrm{e}}^{-\frac{16x^5+12x^3-108x+27}{9(x^2+3)x}}}{x}$	$33$
norman	$\frac{x^3{\mathrm{e}}^{\frac{-16x^5-12x^3+108x-27}{9x^3+27x}}+3x{\mathrm{e}}^{\frac{-16x^5-12x^3+108x-27}{9x^3+27x}}}{x^2(x^2+3)}$	$76$

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 $$\begin{array}{c}\frac{e^{(-\frac{16}{9}{x}^2+\frac{x}{x^2+3}-\frac{1}{x}+4)}}{x}\end{array}$$

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 $$\begin{array}{c}\frac{{\mathrm{e}}^{-\frac{3}{x^3+3x}}{\mathrm{e}}^{-\frac{4x^2}{3(x^2+3)}}{\mathrm{e}}^{-\frac{16x^4}{9(x^2+3)}}{\mathrm{e}}^{\frac{12}{x^2+3}}}{x}\end{array}$$

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 $$\begin{array}{c}\frac{e^{\frac{-16x^5-12x^3+108x-27}{9x^3+27x}}}{x}\end{array}$$

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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