3.70.19 \(\int \frac {e^{5 e^{5+x}}+e^{4 e^{5+x}} (7 x^2-4 e^{5+x} x^3)}{e^{5 e^{5+x}}-5 e^{4 e^{5+x}} x^2+10 e^{3 e^{5+x}} x^4-10 e^{2 e^{5+x}} x^6+5 e^{e^{5+x}} x^8-x^{10}} \, dx\)

Optimal. Leaf size=19 \[ \frac {x}{\left (-1+e^{-e^{5+x}} x^2\right )^4} \]

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Rubi [F]  time = 2.33, 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^{5 e^{5+x}}+e^{4 e^{5+x}} \left (7 x^2-4 e^{5+x} x^3\right )}{e^{5 e^{5+x}}-5 e^{4 e^{5+x}} x^2+10 e^{3 e^{5+x}} x^4-10 e^{2 e^{5+x}} x^6+5 e^{e^{5+x}} x^8-x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5*E^(5 + x)) + E^(4*E^(5 + x))*(7*x^2 - 4*E^(5 + x)*x^3))/(E^(5*E^(5 + x)) - 5*E^(4*E^(5 + x))*x^2 + 1
0*E^(3*E^(5 + x))*x^4 - 10*E^(2*E^(5 + x))*x^6 + 5*E^E^(5 + x)*x^8 - x^10),x]

[Out]

Defer[Int][E^(4*E^(5 + x))/(E^E^(5 + x) - x^2)^4, x] - 8*Defer[Int][(E^(4*E^(5 + x))*x^2)/(-E^E^(5 + x) + x^2)
^5, x] + 4*Defer[Int][(E^(5 + 4*E^(5 + x) + x)*x^3)/(-E^E^(5 + x) + x^2)^5, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 e^{5+x}} \left (e^{e^{5+x}}+7 x^2-4 e^{5+x} x^3\right )}{\left (e^{e^{5+x}}-x^2\right )^5} \, dx\\ &=\int \left (\frac {4 e^{5+4 e^{5+x}+x} x^3}{\left (-e^{e^{5+x}}+x^2\right )^5}+\frac {e^{4 e^{5+x}} \left (e^{e^{5+x}}+7 x^2\right )}{\left (e^{e^{5+x}}-x^2\right )^5}\right ) \, dx\\ &=4 \int \frac {e^{5+4 e^{5+x}+x} x^3}{\left (-e^{e^{5+x}}+x^2\right )^5} \, dx+\int \frac {e^{4 e^{5+x}} \left (e^{e^{5+x}}+7 x^2\right )}{\left (e^{e^{5+x}}-x^2\right )^5} \, dx\\ &=4 \int \frac {e^{5+4 e^{5+x}+x} x^3}{\left (-e^{e^{5+x}}+x^2\right )^5} \, dx+\int \left (\frac {e^{4 e^{5+x}}}{\left (e^{e^{5+x}}-x^2\right )^4}-\frac {8 e^{4 e^{5+x}} x^2}{\left (-e^{e^{5+x}}+x^2\right )^5}\right ) \, dx\\ &=4 \int \frac {e^{5+4 e^{5+x}+x} x^3}{\left (-e^{e^{5+x}}+x^2\right )^5} \, dx-8 \int \frac {e^{4 e^{5+x}} x^2}{\left (-e^{e^{5+x}}+x^2\right )^5} \, dx+\int \frac {e^{4 e^{5+x}}}{\left (e^{e^{5+x}}-x^2\right )^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 26, normalized size = 1.37 \begin {gather*} \frac {e^{4 e^{5+x}} x}{\left (e^{e^{5+x}}-x^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5*E^(5 + x)) + E^(4*E^(5 + x))*(7*x^2 - 4*E^(5 + x)*x^3))/(E^(5*E^(5 + x)) - 5*E^(4*E^(5 + x))*x
^2 + 10*E^(3*E^(5 + x))*x^4 - 10*E^(2*E^(5 + x))*x^6 + 5*E^E^(5 + x)*x^8 - x^10),x]

[Out]

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

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fricas [B]  time = 0.69, size = 56, normalized size = 2.95 \begin {gather*} \frac {x e^{\left (4 \, e^{\left (x + 5\right )}\right )}}{x^{8} - 4 \, x^{6} e^{\left (e^{\left (x + 5\right )}\right )} + 6 \, x^{4} e^{\left (2 \, e^{\left (x + 5\right )}\right )} - 4 \, x^{2} e^{\left (3 \, e^{\left (x + 5\right )}\right )} + e^{\left (4 \, e^{\left (x + 5\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(5)*exp(x))^5+(-4*x^3*exp(5)*exp(x)+7*x^2)*exp(exp(5)*exp(x))^4)/(exp(exp(5)*exp(x))^5-5*x^2
*exp(exp(5)*exp(x))^4+10*x^4*exp(exp(5)*exp(x))^3-10*x^6*exp(exp(5)*exp(x))^2+5*x^8*exp(exp(5)*exp(x))-x^10),x
, algorithm="fricas")

[Out]

x*e^(4*e^(x + 5))/(x^8 - 4*x^6*e^(e^(x + 5)) + 6*x^4*e^(2*e^(x + 5)) - 4*x^2*e^(3*e^(x + 5)) + e^(4*e^(x + 5))
)

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giac [B]  time = 0.35, size = 56, normalized size = 2.95 \begin {gather*} \frac {x e^{\left (4 \, e^{\left (x + 5\right )}\right )}}{x^{8} - 4 \, x^{6} e^{\left (e^{\left (x + 5\right )}\right )} + 6 \, x^{4} e^{\left (2 \, e^{\left (x + 5\right )}\right )} - 4 \, x^{2} e^{\left (3 \, e^{\left (x + 5\right )}\right )} + e^{\left (4 \, e^{\left (x + 5\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(5)*exp(x))^5+(-4*x^3*exp(5)*exp(x)+7*x^2)*exp(exp(5)*exp(x))^4)/(exp(exp(5)*exp(x))^5-5*x^2
*exp(exp(5)*exp(x))^4+10*x^4*exp(exp(5)*exp(x))^3-10*x^6*exp(exp(5)*exp(x))^2+5*x^8*exp(exp(5)*exp(x))-x^10),x
, algorithm="giac")

[Out]

x*e^(4*e^(x + 5))/(x^8 - 4*x^6*e^(e^(x + 5)) + 6*x^4*e^(2*e^(x + 5)) - 4*x^2*e^(3*e^(x + 5)) + e^(4*e^(x + 5))
)

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maple [B]  time = 0.06, size = 56, normalized size = 2.95

method	result	size
risch	\(x -\frac {\left (x^{6}-4 x^{4} {\mathrm e}^{{\mathrm e}^{5+x}}+6 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{5+x}}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{5+x}}\right ) x^{3}}{\left (x^{2}-{\mathrm e}^{{\mathrm e}^{5+x}}\right )^{4}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(5)*exp(x))^5+(-4*x^3*exp(5)*exp(x)+7*x^2)*exp(exp(5)*exp(x))^4)/(exp(exp(5)*exp(x))^5-5*x^2*exp(e
xp(5)*exp(x))^4+10*x^4*exp(exp(5)*exp(x))^3-10*x^6*exp(exp(5)*exp(x))^2+5*x^8*exp(exp(5)*exp(x))-x^10),x,metho
d=_RETURNVERBOSE)

[Out]

x-(x^6-4*x^4*exp(exp(5+x))+6*x^2*exp(2*exp(5+x))-4*exp(3*exp(5+x)))*x^3/(x^2-exp(exp(5+x)))^4

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maxima [B]  time = 0.74, size = 56, normalized size = 2.95 \begin {gather*} \frac {x e^{\left (4 \, e^{\left (x + 5\right )}\right )}}{x^{8} - 4 \, x^{6} e^{\left (e^{\left (x + 5\right )}\right )} + 6 \, x^{4} e^{\left (2 \, e^{\left (x + 5\right )}\right )} - 4 \, x^{2} e^{\left (3 \, e^{\left (x + 5\right )}\right )} + e^{\left (4 \, e^{\left (x + 5\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(5)*exp(x))^5+(-4*x^3*exp(5)*exp(x)+7*x^2)*exp(exp(5)*exp(x))^4)/(exp(exp(5)*exp(x))^5-5*x^2
*exp(exp(5)*exp(x))^4+10*x^4*exp(exp(5)*exp(x))^3-10*x^6*exp(exp(5)*exp(x))^2+5*x^8*exp(exp(5)*exp(x))-x^10),x
, algorithm="maxima")

[Out]

x*e^(4*e^(x + 5))/(x^8 - 4*x^6*e^(e^(x + 5)) + 6*x^4*e^(2*e^(x + 5)) - 4*x^2*e^(3*e^(x + 5)) + e^(4*e^(x + 5))
)

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mupad [B]  time = 4.38, size = 250, normalized size = 13.16 \begin {gather*} x-\frac {6\,\left (x^7\,{\mathrm {e}}^{x+5}-2\,x^6\right )}{\left (2\,x-x^2\,{\mathrm {e}}^{x+5}\right )\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^5\,{\mathrm {e}}^x}-2\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^5\,{\mathrm {e}}^x}+x^4\right )}-\frac {4\,\left (x^5\,{\mathrm {e}}^{x+5}-2\,x^4\right )}{\left ({\mathrm {e}}^{{\mathrm {e}}^5\,{\mathrm {e}}^x}-x^2\right )\,\left (2\,x-x^2\,{\mathrm {e}}^{x+5}\right )}-\frac {x^{11}\,{\mathrm {e}}^{x+5}-2\,x^{10}}{\left (2\,x-x^2\,{\mathrm {e}}^{x+5}\right )\,\left ({\mathrm {e}}^{4\,{\mathrm {e}}^5\,{\mathrm {e}}^x}-4\,x^2\,{\mathrm {e}}^{3\,{\mathrm {e}}^5\,{\mathrm {e}}^x}+6\,x^4\,{\mathrm {e}}^{2\,{\mathrm {e}}^5\,{\mathrm {e}}^x}-4\,x^6\,{\mathrm {e}}^{{\mathrm {e}}^5\,{\mathrm {e}}^x}+x^8\right )}-\frac {4\,\left (x^9\,{\mathrm {e}}^{x+5}-2\,x^8\right )}{\left (2\,x-x^2\,{\mathrm {e}}^{x+5}\right )\,\left ({\mathrm {e}}^{3\,{\mathrm {e}}^5\,{\mathrm {e}}^x}-3\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^5\,{\mathrm {e}}^x}+3\,x^4\,{\mathrm {e}}^{{\mathrm {e}}^5\,{\mathrm {e}}^x}-x^6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5*exp(5)*exp(x)) + exp(4*exp(5)*exp(x))*(7*x^2 - 4*x^3*exp(5)*exp(x)))/(exp(5*exp(5)*exp(x)) - 5*x^2*
exp(4*exp(5)*exp(x)) + 10*x^4*exp(3*exp(5)*exp(x)) - 10*x^6*exp(2*exp(5)*exp(x)) + 5*x^8*exp(exp(5)*exp(x)) -
x^10),x)

[Out]

x - (6*(x^7*exp(x + 5) - 2*x^6))/((2*x - x^2*exp(x + 5))*(exp(2*exp(5)*exp(x)) - 2*x^2*exp(exp(5)*exp(x)) + x^
4)) - (4*(x^5*exp(x + 5) - 2*x^4))/((exp(exp(5)*exp(x)) - x^2)*(2*x - x^2*exp(x + 5))) - (x^11*exp(x + 5) - 2*
x^10)/((2*x - x^2*exp(x + 5))*(exp(4*exp(5)*exp(x)) - 4*x^2*exp(3*exp(5)*exp(x)) + 6*x^4*exp(2*exp(5)*exp(x))
- 4*x^6*exp(exp(5)*exp(x)) + x^8)) - (4*(x^9*exp(x + 5) - 2*x^8))/((2*x - x^2*exp(x + 5))*(exp(3*exp(5)*exp(x)
) - 3*x^2*exp(2*exp(5)*exp(x)) + 3*x^4*exp(exp(5)*exp(x)) - x^6))

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sympy [B]  time = 0.24, size = 105, normalized size = 5.53 \begin {gather*} x + \frac {- x^{9} + 4 x^{7} e^{e^{5} e^{x}} - 6 x^{5} e^{2 e^{5} e^{x}} + 4 x^{3} e^{3 e^{5} e^{x}}}{x^{8} - 4 x^{6} e^{e^{5} e^{x}} + 6 x^{4} e^{2 e^{5} e^{x}} - 4 x^{2} e^{3 e^{5} e^{x}} + e^{4 e^{5} e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(exp(5)*exp(x))**5+(-4*x**3*exp(5)*exp(x)+7*x**2)*exp(exp(5)*exp(x))**4)/(exp(exp(5)*exp(x))**5-
5*x**2*exp(exp(5)*exp(x))**4+10*x**4*exp(exp(5)*exp(x))**3-10*x**6*exp(exp(5)*exp(x))**2+5*x**8*exp(exp(5)*exp
(x))-x**10),x)

[Out]

x + (-x**9 + 4*x**7*exp(exp(5)*exp(x)) - 6*x**5*exp(2*exp(5)*exp(x)) + 4*x**3*exp(3*exp(5)*exp(x)))/(x**8 - 4*
x**6*exp(exp(5)*exp(x)) + 6*x**4*exp(2*exp(5)*exp(x)) - 4*x**2*exp(3*exp(5)*exp(x)) + exp(4*exp(5)*exp(x)))

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