3.49.81 \(\int \frac {e^{\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}} (352+44 x-33 x^2)}{576 x^2+144 x^3+9 x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{3} e^{\frac {11 e^{3-\frac {4}{3 x}}}{8+x}} \]

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Rubi [F]  time = 1.87, 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 {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right ) \left (352+44 x-33 x^2\right )}{576 x^2+144 x^3+9 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((11*E^((-4 + 9*x)/(3*x)))/(8 + x) + (-4 + 9*x)/(3*x))*(352 + 44*x - 33*x^2))/(576*x^2 + 144*x^3 + 9*x^
4),x]

[Out]

(11*Defer[Int][E^((11*E^((-4 + 9*x)/(3*x)))/(8 + x) + (-4 + 9*x)/(3*x))/x^2, x])/18 - (11*Defer[Int][E^((11*E^
((-4 + 9*x)/(3*x)))/(8 + x) + (-4 + 9*x)/(3*x))/x, x])/144 - (11*Defer[Int][E^((11*E^((-4 + 9*x)/(3*x)))/(8 +
x) + (-4 + 9*x)/(3*x))/(8 + x)^2, x])/3 + (11*Defer[Int][E^((11*E^((-4 + 9*x)/(3*x)))/(8 + x) + (-4 + 9*x)/(3*
x))/(8 + x), x])/144

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right ) \left (352+44 x-33 x^2\right )}{x^2 \left (576+144 x+9 x^2\right )} \, dx\\ &=\int \frac {\exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right ) \left (352+44 x-33 x^2\right )}{9 x^2 (8+x)^2} \, dx\\ &=\frac {1}{9} \int \frac {\exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right ) \left (352+44 x-33 x^2\right )}{x^2 (8+x)^2} \, dx\\ &=\frac {1}{9} \int \left (\frac {11 \exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{2 x^2}-\frac {11 \exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{16 x}-\frac {33 \exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{(8+x)^2}+\frac {11 \exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{16 (8+x)}\right ) \, dx\\ &=-\left (\frac {11}{144} \int \frac {\exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{x} \, dx\right )+\frac {11}{144} \int \frac {\exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{8+x} \, dx+\frac {11}{18} \int \frac {\exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{x^2} \, dx-\frac {11}{3} \int \frac {\exp \left (\frac {11 e^{\frac {-4+9 x}{3 x}}}{8+x}+\frac {-4+9 x}{3 x}\right )}{(8+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{3} e^{\frac {11 e^{3-\frac {4}{3 x}}}{8+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((11*E^((-4 + 9*x)/(3*x)))/(8 + x) + (-4 + 9*x)/(3*x))*(352 + 44*x - 33*x^2))/(576*x^2 + 144*x^3
+ 9*x^4),x]

[Out]

E^((11*E^(3 - 4/(3*x)))/(8 + x))/3

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fricas [B]  time = 0.75, size = 49, normalized size = 2.04 \begin {gather*} \frac {1}{3} \, e^{\left (\frac {9 \, x^{2} + 33 \, x e^{\left (\frac {9 \, x - 4}{3 \, x}\right )} + 68 \, x - 32}{3 \, {\left (x^{2} + 8 \, x\right )}} - \frac {9 \, x - 4}{3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-33*x^2+44*x+352)*exp(1/3*(9*x-4)/x)*exp(11*exp(1/3*(9*x-4)/x)/(x+8))/(9*x^4+144*x^3+576*x^2),x, al
gorithm="fricas")

[Out]

1/3*e^(1/3*(9*x^2 + 33*x*e^(1/3*(9*x - 4)/x) + 68*x - 32)/(x^2 + 8*x) - 1/3*(9*x - 4)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {11 \, {\left (3 \, x^{2} - 4 \, x - 32\right )} e^{\left (\frac {9 \, x - 4}{3 \, x} + \frac {11 \, e^{\left (\frac {9 \, x - 4}{3 \, x}\right )}}{x + 8}\right )}}{9 \, {\left (x^{4} + 16 \, x^{3} + 64 \, x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-33*x^2+44*x+352)*exp(1/3*(9*x-4)/x)*exp(11*exp(1/3*(9*x-4)/x)/(x+8))/(9*x^4+144*x^3+576*x^2),x, al
gorithm="giac")

[Out]

integrate(-11/9*(3*x^2 - 4*x - 32)*e^(1/3*(9*x - 4)/x + 11*e^(1/3*(9*x - 4)/x)/(x + 8))/(x^4 + 16*x^3 + 64*x^2
), x)

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maple [A]  time = 0.10, size = 22, normalized size = 0.92

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {11 \,{\mathrm e}^{\frac {9 x -4}{3 x}}}{x +8}}}{3}\)	\(22\)
norman	\(\frac {\frac {8 x \,{\mathrm e}^{\frac {11 \,{\mathrm e}^{\frac {9 x -4}{3 x}}}{x +8}}}{3}+\frac {x^{2} {\mathrm e}^{\frac {11 \,{\mathrm e}^{\frac {9 x -4}{3 x}}}{x +8}}}{3}}{x \left (x +8\right )}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-33*x^2+44*x+352)*exp(1/3*(9*x-4)/x)*exp(11*exp(1/3*(9*x-4)/x)/(x+8))/(9*x^4+144*x^3+576*x^2),x,method=_R
ETURNVERBOSE)

[Out]

1/3*exp(11*exp(1/3*(9*x-4)/x)/(x+8))

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maxima [A]  time = 0.53, size = 18, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, e^{\left (\frac {11 \, e^{\left (-\frac {4}{3 \, x} + 3\right )}}{x + 8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-33*x^2+44*x+352)*exp(1/3*(9*x-4)/x)*exp(11*exp(1/3*(9*x-4)/x)/(x+8))/(9*x^4+144*x^3+576*x^2),x, al
gorithm="maxima")

[Out]

1/3*e^(11*e^(-4/3/x + 3)/(x + 8))

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mupad [B]  time = 3.67, size = 18, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {e}}^{\frac {11\,{\mathrm {e}}^3\,{\mathrm {e}}^{-\frac {4}{3\,x}}}{x+8}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((11*exp((3*x - 4/3)/x))/(x + 8))*exp((3*x - 4/3)/x)*(44*x - 33*x^2 + 352))/(576*x^2 + 144*x^3 + 9*x^4
),x)

[Out]

exp((11*exp(3)*exp(-4/(3*x)))/(x + 8))/3

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sympy [A]  time = 0.58, size = 17, normalized size = 0.71 \begin {gather*} \frac {e^{\frac {11 e^{\frac {3 x - \frac {4}{3}}{x}}}{x + 8}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-33*x**2+44*x+352)*exp(1/3*(9*x-4)/x)*exp(11*exp(1/3*(9*x-4)/x)/(x+8))/(9*x**4+144*x**3+576*x**2),x
)

[Out]

exp(11*exp((3*x - 4/3)/x)/(x + 8))/3

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