3.104.35 \(\int \frac {162 x^5+9 x^6+e^{2 e^x} (135 x^4-108 e^x x^5)}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} (162 x+27 x^2)+e^{2 e^x} (324 x^2+108 x^3+9 x^4)} \, dx\)

Optimal. Leaf size=31 \[ \frac {x^3}{\left (\frac {x}{3}+\left (\frac {e^{2 e^x}}{x^2}+\frac {2}{x}\right ) x\right )^2} \]

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Rubi [F]  time = 2.49, 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 {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(162*x^5 + 9*x^6 + E^(2*E^x)*(135*x^4 - 108*E^x*x^5))/(27*E^(6*E^x) + 216*x^3 + 108*x^4 + 18*x^5 + x^6 + E
^(4*E^x)*(162*x + 27*x^2) + E^(2*E^x)*(324*x^2 + 108*x^3 + 9*x^4)),x]

[Out]

-108*Defer[Int][x^5/(3*E^(2*E^x) + 6*x + x^2)^3, x] - 108*Defer[Int][(E^(2*E^x + x)*x^5)/(3*E^(2*E^x) + 6*x +
x^2)^3, x] - 36*Defer[Int][x^6/(3*E^(2*E^x) + 6*x + x^2)^3, x] + 45*Defer[Int][x^4/(3*E^(2*E^x) + 6*x + x^2)^2
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 x^4 \left (15 e^{2 e^x}-12 e^{2 e^x+x} x+x (18+x)\right )}{\left (3 e^{2 e^x}+x (6+x)\right )^3} \, dx\\ &=9 \int \frac {x^4 \left (15 e^{2 e^x}-12 e^{2 e^x+x} x+x (18+x)\right )}{\left (3 e^{2 e^x}+x (6+x)\right )^3} \, dx\\ &=9 \int \left (-\frac {12 e^{2 e^x+x} x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3}+\frac {x^4 \left (15 e^{2 e^x}+18 x+x^2\right )}{\left (3 e^{2 e^x}+6 x+x^2\right )^3}\right ) \, dx\\ &=9 \int \frac {x^4 \left (15 e^{2 e^x}+18 x+x^2\right )}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx-108 \int \frac {e^{2 e^x+x} x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx\\ &=9 \int \left (-\frac {4 x^5 (3+x)}{\left (3 e^{2 e^x}+6 x+x^2\right )^3}+\frac {5 x^4}{\left (3 e^{2 e^x}+6 x+x^2\right )^2}\right ) \, dx-108 \int \frac {e^{2 e^x+x} x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx\\ &=-\left (36 \int \frac {x^5 (3+x)}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx\right )+45 \int \frac {x^4}{\left (3 e^{2 e^x}+6 x+x^2\right )^2} \, dx-108 \int \frac {e^{2 e^x+x} x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx\\ &=-\left (36 \int \left (\frac {3 x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3}+\frac {x^6}{\left (3 e^{2 e^x}+6 x+x^2\right )^3}\right ) \, dx\right )+45 \int \frac {x^4}{\left (3 e^{2 e^x}+6 x+x^2\right )^2} \, dx-108 \int \frac {e^{2 e^x+x} x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx\\ &=-\left (36 \int \frac {x^6}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx\right )+45 \int \frac {x^4}{\left (3 e^{2 e^x}+6 x+x^2\right )^2} \, dx-108 \int \frac {x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx-108 \int \frac {e^{2 e^x+x} x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 23, normalized size = 0.74 \begin {gather*} \frac {9 x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(162*x^5 + 9*x^6 + E^(2*E^x)*(135*x^4 - 108*E^x*x^5))/(27*E^(6*E^x) + 216*x^3 + 108*x^4 + 18*x^5 + x
^6 + E^(4*E^x)*(162*x + 27*x^2) + E^(2*E^x)*(324*x^2 + 108*x^3 + 9*x^4)),x]

[Out]

(9*x^5)/(3*E^(2*E^x) + 6*x + x^2)^2

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fricas [A]  time = 0.77, size = 42, normalized size = 1.35 \begin {gather*} \frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 36 \, x^{2} + 6 \, {\left (x^{2} + 6 \, x\right )} e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp(exp(x))^6+(27*x^2+162*x)*exp(exp(x))
^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x))^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm="fricas")

[Out]

9*x^5/(x^4 + 12*x^3 + 36*x^2 + 6*(x^2 + 6*x)*e^(2*e^x) + 9*e^(4*e^x))

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giac [A]  time = 0.37, size = 46, normalized size = 1.48 \begin {gather*} \frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 6 \, x^{2} e^{\left (2 \, e^{x}\right )} + 36 \, x^{2} + 36 \, x e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp(exp(x))^6+(27*x^2+162*x)*exp(exp(x))
^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x))^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm="giac")

[Out]

9*x^5/(x^4 + 12*x^3 + 6*x^2*e^(2*e^x) + 36*x^2 + 36*x*e^(2*e^x) + 9*e^(4*e^x))

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maple [A]  time = 0.05, size = 22, normalized size = 0.71

method	result	size
risch	\(\frac {9 x^{5}}{\left (3 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+x^{2}+6 x \right )^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp(exp(x))^6+(27*x^2+162*x)*exp(exp(x))^4+(9*
x^4+108*x^3+324*x^2)*exp(exp(x))^2+x^6+18*x^5+108*x^4+216*x^3),x,method=_RETURNVERBOSE)

[Out]

9*x^5/(3*exp(2*exp(x))+x^2+6*x)^2

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maxima [A]  time = 0.43, size = 42, normalized size = 1.35 \begin {gather*} \frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 36 \, x^{2} + 6 \, {\left (x^{2} + 6 \, x\right )} e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp(exp(x))^6+(27*x^2+162*x)*exp(exp(x))
^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x))^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm="maxima")

[Out]

9*x^5/(x^4 + 12*x^3 + 36*x^2 + 6*(x^2 + 6*x)*e^(2*e^x) + 9*e^(4*e^x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {162\,x^5-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (108\,x^5\,{\mathrm {e}}^x-135\,x^4\right )+9\,x^6}{27\,{\mathrm {e}}^{6\,{\mathrm {e}}^x}+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (9\,x^4+108\,x^3+324\,x^2\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^x}\,\left (27\,x^2+162\,x\right )+216\,x^3+108\,x^4+18\,x^5+x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((162*x^5 - exp(2*exp(x))*(108*x^5*exp(x) - 135*x^4) + 9*x^6)/(27*exp(6*exp(x)) + exp(2*exp(x))*(324*x^2 +
108*x^3 + 9*x^4) + exp(4*exp(x))*(162*x + 27*x^2) + 216*x^3 + 108*x^4 + 18*x^5 + x^6),x)

[Out]

int((162*x^5 - exp(2*exp(x))*(108*x^5*exp(x) - 135*x^4) + 9*x^6)/(27*exp(6*exp(x)) + exp(2*exp(x))*(324*x^2 +
108*x^3 + 9*x^4) + exp(4*exp(x))*(162*x + 27*x^2) + 216*x^3 + 108*x^4 + 18*x^5 + x^6), x)

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sympy [A]  time = 0.21, size = 41, normalized size = 1.32 \begin {gather*} \frac {9 x^{5}}{x^{4} + 12 x^{3} + 36 x^{2} + \left (6 x^{2} + 36 x\right ) e^{2 e^{x}} + 9 e^{4 e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-108*x**5*exp(x)+135*x**4)*exp(exp(x))**2+9*x**6+162*x**5)/(27*exp(exp(x))**6+(27*x**2+162*x)*exp(
exp(x))**4+(9*x**4+108*x**3+324*x**2)*exp(exp(x))**2+x**6+18*x**5+108*x**4+216*x**3),x)

[Out]

9*x**5/(x**4 + 12*x**3 + 36*x**2 + (6*x**2 + 36*x)*exp(2*exp(x)) + 9*exp(4*exp(x)))

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