3.54.47 \(\int \frac {20 x^4-10 x^5+e^{e^x} (e^{2 x} (-16 x^3+8 x^4-x^5)+e^x (48 x^2-40 x^3+11 x^4-x^5))}{25 x^4+e^{2 e^x+2 x} (16-8 x+x^2)+e^{e^x+x} (40 x^2-10 x^3)} \, dx\)

Optimal. Leaf size=25 \[ -2+\frac {x}{-\frac {5}{-4+x}+\frac {e^{e^x+x}}{x^2}} \]

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Rubi [F]  time = 6.56, 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 {20 x^4-10 x^5+e^{e^x} \left (e^{2 x} \left (-16 x^3+8 x^4-x^5\right )+e^x \left (48 x^2-40 x^3+11 x^4-x^5\right )\right )}{25 x^4+e^{2 e^x+2 x} \left (16-8 x+x^2\right )+e^{e^x+x} \left (40 x^2-10 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20*x^4 - 10*x^5 + E^E^x*(E^(2*x)*(-16*x^3 + 8*x^4 - x^5) + E^x*(48*x^2 - 40*x^3 + 11*x^4 - x^5)))/(25*x^4
+ E^(2*E^x + 2*x)*(16 - 8*x + x^2) + E^(E^x + x)*(40*x^2 - 10*x^3)),x]

[Out]

-Defer[Int][x^3/E^E^x, x] - 40*Defer[Int][x^4/(E^(E^x + x)*(-4 + x) - 5*x^2)^2, x] + 25*Defer[Int][x^5/(E^(E^x
+ x)*(-4 + x) - 5*x^2)^2, x] - 5*Defer[Int][x^6/(E^(E^x + x)*(-4 + x) - 5*x^2)^2, x] - 25*Defer[Int][x^7/(E^E
^x*(E^(E^x + x)*(-4 + x) - 5*x^2)^2), x] - 12*Defer[Int][x^2/(E^(E^x + x)*(-4 + x) - 5*x^2), x] + 7*Defer[Int]
[x^3/(E^(E^x + x)*(-4 + x) - 5*x^2), x] - Defer[Int][x^4/(E^(E^x + x)*(-4 + x) - 5*x^2), x] + 10*Defer[Int][x^
5/(E^E^x*(-(E^(E^x + x)*(-4 + x)) + 5*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (-e^{e^x+x} (-4+x)^2 (-3+x)-e^{e^x+2 x} (-4+x)^2 x-10 (-2+x) x^2\right )}{\left (e^{e^x+x} (-4+x)-5 x^2\right )^2} \, dx\\ &=\int \left (-e^{-e^x} x^3-\frac {5 e^{-e^x} x^4 \left (8 e^{e^x}-5 e^{e^x} x+e^{e^x} x^2+5 x^3\right )}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2}-\frac {e^{-e^x} x^2 \left (12 e^{e^x}-7 e^{e^x} x+e^{e^x} x^2+10 x^3\right )}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2}\right ) \, dx\\ &=-\left (5 \int \frac {e^{-e^x} x^4 \left (8 e^{e^x}-5 e^{e^x} x+e^{e^x} x^2+5 x^3\right )}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2} \, dx\right )-\int e^{-e^x} x^3 \, dx-\int \frac {e^{-e^x} x^2 \left (12 e^{e^x}-7 e^{e^x} x+e^{e^x} x^2+10 x^3\right )}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2} \, dx\\ &=-\left (5 \int \frac {e^{-e^x} \left (5 x^7+e^{e^x} x^4 \left (8-5 x+x^2\right )\right )}{\left (e^{e^x+x} (-4+x)-5 x^2\right )^2} \, dx\right )-\int e^{-e^x} x^3 \, dx-\int \frac {e^{-e^x} \left (10 x^5+e^{e^x} x^2 \left (12-7 x+x^2\right )\right )}{e^{e^x+x} (-4+x)-5 x^2} \, dx\\ &=-\left (5 \int \left (\frac {8 x^4}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2}-\frac {5 x^5}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2}+\frac {x^6}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2}+\frac {5 e^{-e^x} x^7}{\left (4 e^{e^x+x}-e^{e^x+x} x+5 x^2\right )^2}\right ) \, dx\right )-\int e^{-e^x} x^3 \, dx-\int \left (\frac {12 x^2}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2}-\frac {7 x^3}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2}+\frac {x^4}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2}-\frac {10 e^{-e^x} x^5}{4 e^{e^x+x}-e^{e^x+x} x+5 x^2}\right ) \, dx\\ &=-\left (5 \int \frac {x^6}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2} \, dx\right )+7 \int \frac {x^3}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2} \, dx+10 \int \frac {e^{-e^x} x^5}{4 e^{e^x+x}-e^{e^x+x} x+5 x^2} \, dx-12 \int \frac {x^2}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2} \, dx+25 \int \frac {x^5}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2} \, dx-25 \int \frac {e^{-e^x} x^7}{\left (4 e^{e^x+x}-e^{e^x+x} x+5 x^2\right )^2} \, dx-40 \int \frac {x^4}{\left (-4 e^{e^x+x}+e^{e^x+x} x-5 x^2\right )^2} \, dx-\int e^{-e^x} x^3 \, dx-\int \frac {x^4}{-4 e^{e^x+x}+e^{e^x+x} x-5 x^2} \, dx\\ &=-\left (5 \int \frac {x^6}{\left (e^{e^x+x} (-4+x)-5 x^2\right )^2} \, dx\right )+7 \int \frac {x^3}{e^{e^x+x} (-4+x)-5 x^2} \, dx+10 \int \frac {e^{-e^x} x^5}{-e^{e^x+x} (-4+x)+5 x^2} \, dx-12 \int \frac {x^2}{e^{e^x+x} (-4+x)-5 x^2} \, dx+25 \int \frac {x^5}{\left (e^{e^x+x} (-4+x)-5 x^2\right )^2} \, dx-25 \int \frac {e^{-e^x} x^7}{\left (e^{e^x+x} (-4+x)-5 x^2\right )^2} \, dx-40 \int \frac {x^4}{\left (e^{e^x+x} (-4+x)-5 x^2\right )^2} \, dx-\int e^{-e^x} x^3 \, dx-\int \frac {x^4}{e^{e^x+x} (-4+x)-5 x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.88, size = 28, normalized size = 1.12 \begin {gather*} -\frac {(-4+x) x^3}{-e^{e^x+x} (-4+x)+5 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*x^4 - 10*x^5 + E^E^x*(E^(2*x)*(-16*x^3 + 8*x^4 - x^5) + E^x*(48*x^2 - 40*x^3 + 11*x^4 - x^5)))/(
25*x^4 + E^(2*E^x + 2*x)*(16 - 8*x + x^2) + E^(E^x + x)*(40*x^2 - 10*x^3)),x]

[Out]

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

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fricas [A]  time = 0.62, size = 29, normalized size = 1.16 \begin {gather*} -\frac {x^{4} - 4 \, x^{3}}{5 \, x^{2} - {\left (x - 4\right )} e^{\left (x + e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^5+8*x^4-16*x^3)*exp(x)^2+(-x^5+11*x^4-40*x^3+48*x^2)*exp(x))*exp(exp(x))-10*x^5+20*x^4)/((x^2-
8*x+16)*exp(x)^2*exp(exp(x))^2+(-10*x^3+40*x^2)*exp(x)*exp(exp(x))+25*x^4),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.25, size = 34, normalized size = 1.36 \begin {gather*} -\frac {x^{4} - 4 \, x^{3}}{5 \, x^{2} - x e^{\left (x + e^{x}\right )} + 4 \, e^{\left (x + e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^5+8*x^4-16*x^3)*exp(x)^2+(-x^5+11*x^4-40*x^3+48*x^2)*exp(x))*exp(exp(x))-10*x^5+20*x^4)/((x^2-
8*x+16)*exp(x)^2*exp(exp(x))^2+(-10*x^3+40*x^2)*exp(x)*exp(exp(x))+25*x^4),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 32, normalized size = 1.28

method	result	size
risch	\(-\frac {\left (x -4\right ) x^{3}}{-x \,{\mathrm e}^{{\mathrm e}^{x}+x}+5 x^{2}+4 \,{\mathrm e}^{{\mathrm e}^{x}+x}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^5+8*x^4-16*x^3)*exp(x)^2+(-x^5+11*x^4-40*x^3+48*x^2)*exp(x))*exp(exp(x))-10*x^5+20*x^4)/((x^2-8*x+16
)*exp(x)^2*exp(exp(x))^2+(-10*x^3+40*x^2)*exp(x)*exp(exp(x))+25*x^4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.53, size = 29, normalized size = 1.16 \begin {gather*} -\frac {x^{4} - 4 \, x^{3}}{5 \, x^{2} - {\left (x - 4\right )} e^{\left (x + e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^5+8*x^4-16*x^3)*exp(x)^2+(-x^5+11*x^4-40*x^3+48*x^2)*exp(x))*exp(exp(x))-10*x^5+20*x^4)/((x^2-
8*x+16)*exp(x)^2*exp(exp(x))^2+(-10*x^3+40*x^2)*exp(x)*exp(exp(x))+25*x^4),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{2\,x}\,\left (x^5-8\,x^4+16\,x^3\right )-{\mathrm {e}}^x\,\left (-x^5+11\,x^4-40\,x^3+48\,x^2\right )\right )-20\,x^4+10\,x^5}{25\,x^4+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (x^2-8\,x+16\right )+{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^x\,\left (40\,x^2-10\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x))*(exp(2*x)*(16*x^3 - 8*x^4 + x^5) - exp(x)*(48*x^2 - 40*x^3 + 11*x^4 - x^5)) - 20*x^4 + 10*x^
5)/(25*x^4 + exp(2*x)*exp(2*exp(x))*(x^2 - 8*x + 16) + exp(exp(x))*exp(x)*(40*x^2 - 10*x^3)),x)

[Out]

int(-(exp(exp(x))*(exp(2*x)*(16*x^3 - 8*x^4 + x^5) - exp(x)*(48*x^2 - 40*x^3 + 11*x^4 - x^5)) - 20*x^4 + 10*x^
5)/(25*x^4 + exp(2*x)*exp(2*exp(x))*(x^2 - 8*x + 16) + exp(exp(x))*exp(x)*(40*x^2 - 10*x^3)), x)

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sympy [A]  time = 0.28, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^{4} - 4 x^{3}}{- 5 x^{2} + \left (x e^{x} - 4 e^{x}\right ) e^{e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**5+8*x**4-16*x**3)*exp(x)**2+(-x**5+11*x**4-40*x**3+48*x**2)*exp(x))*exp(exp(x))-10*x**5+20*x*
*4)/((x**2-8*x+16)*exp(x)**2*exp(exp(x))**2+(-10*x**3+40*x**2)*exp(x)*exp(exp(x))+25*x**4),x)

[Out]

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

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