3.70.46 \(\int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} (e^{e^2} x^3-4 x^4)}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx\)

Optimal. Leaf size=24 \[ \frac {x}{\left (5 e^{2 e^{-e^2} x}+\frac {5}{x}\right )^2} \]

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Rubi [F]  time = 1.92, 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 {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*E^E^2*x^2 + E^((2*x)/E^E^2)*(E^E^2*x^3 - 4*x^4))/(25*E^E^2 + 75*E^(E^2 + (2*x)/E^E^2)*x + 75*E^(E^2 + (
4*x)/E^E^2)*x^2 + 25*E^(E^2 + (6*x)/E^E^2)*x^3),x]

[Out]

(2*Defer[Int][x^2/(1 + E^((2*x)/E^E^2)*x)^3, x])/25 + (4*Defer[Int][x^3/(1 + E^((2*x)/E^E^2)*x)^3, x])/(25*E^E
^2) + Defer[Int][x^2/(1 + E^((2*x)/E^E^2)*x)^2, x]/25 - (4*Defer[Int][x^3/(1 + E^((2*x)/E^E^2)*x)^2, x])/(25*E
^E^2)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-e^2} x^2 \left (3 e^{e^2}+e^{e^2+2 e^{-e^2} x} x-4 e^{2 e^{-e^2} x} x^2\right )}{25 \left (1+e^{2 e^{-e^2} x} x\right )^3} \, dx\\ &=\frac {1}{25} e^{-e^2} \int \frac {x^2 \left (3 e^{e^2}+e^{e^2+2 e^{-e^2} x} x-4 e^{2 e^{-e^2} x} x^2\right )}{\left (1+e^{2 e^{-e^2} x} x\right )^3} \, dx\\ &=\frac {1}{25} e^{-e^2} \int \left (\frac {2 x^2 \left (e^{e^2}+2 x\right )}{\left (1+e^{2 e^{-e^2} x} x\right )^3}+\frac {\left (e^{e^2}-4 x\right ) x^2}{\left (1+e^{2 e^{-e^2} x} x\right )^2}\right ) \, dx\\ &=\frac {1}{25} e^{-e^2} \int \frac {\left (e^{e^2}-4 x\right ) x^2}{\left (1+e^{2 e^{-e^2} x} x\right )^2} \, dx+\frac {1}{25} \left (2 e^{-e^2}\right ) \int \frac {x^2 \left (e^{e^2}+2 x\right )}{\left (1+e^{2 e^{-e^2} x} x\right )^3} \, dx\\ &=\frac {1}{25} e^{-e^2} \int \left (\frac {e^{e^2} x^2}{\left (1+e^{2 e^{-e^2} x} x\right )^2}-\frac {4 x^3}{\left (1+e^{2 e^{-e^2} x} x\right )^2}\right ) \, dx+\frac {1}{25} \left (2 e^{-e^2}\right ) \int \left (\frac {e^{e^2} x^2}{\left (1+e^{2 e^{-e^2} x} x\right )^3}+\frac {2 x^3}{\left (1+e^{2 e^{-e^2} x} x\right )^3}\right ) \, dx\\ &=\frac {1}{25} \int \frac {x^2}{\left (1+e^{2 e^{-e^2} x} x\right )^2} \, dx+\frac {2}{25} \int \frac {x^2}{\left (1+e^{2 e^{-e^2} x} x\right )^3} \, dx+\frac {1}{25} \left (4 e^{-e^2}\right ) \int \frac {x^3}{\left (1+e^{2 e^{-e^2} x} x\right )^3} \, dx-\frac {1}{25} \left (4 e^{-e^2}\right ) \int \frac {x^3}{\left (1+e^{2 e^{-e^2} x} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 25, normalized size = 1.04 \begin {gather*} \frac {x^3}{25 \left (1+e^{2 e^{-e^2} x} x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*E^E^2*x^2 + E^((2*x)/E^E^2)*(E^E^2*x^3 - 4*x^4))/(25*E^E^2 + 75*E^(E^2 + (2*x)/E^E^2)*x + 75*E^(E
^2 + (4*x)/E^E^2)*x^2 + 25*E^(E^2 + (6*x)/E^E^2)*x^3),x]

[Out]

x^3/(25*(1 + E^((2*x)/E^E^2)*x)^2)

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fricas [B]  time = 0.63, size = 61, normalized size = 2.54 \begin {gather*} \frac {x^{3} e^{\left (2 \, e^{2}\right )}}{25 \, {\left (x^{2} e^{\left (2 \, {\left (2 \, x + e^{\left (e^{2} + 2\right )}\right )} e^{\left (-e^{2}\right )}\right )} + 2 \, x e^{\left ({\left (2 \, x + e^{\left (e^{2} + 2\right )}\right )} e^{\left (-e^{2}\right )} + e^{2}\right )} + e^{\left (2 \, e^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2)))/(25*x^3*exp(exp(2))*exp(2*x/exp(exp
(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp(exp(2)))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x, alg
orithm="fricas")

[Out]

1/25*x^3*e^(2*e^2)/(x^2*e^(2*(2*x + e^(e^2 + 2))*e^(-e^2)) + 2*x*e^((2*x + e^(e^2 + 2))*e^(-e^2) + e^2) + e^(2
*e^2))

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giac [B]  time = 0.23, size = 199, normalized size = 8.29 \begin {gather*} \frac {8 \, x^{6} e^{\left (-2 \, e^{2}\right )} + 12 \, x^{5} e^{\left (-e^{2}\right )} + 6 \, x^{4} + x^{3} e^{\left (e^{2}\right )}}{25 \, {\left (8 \, x^{5} e^{\left (4 \, x e^{\left (-e^{2}\right )} - 2 \, e^{2}\right )} + 12 \, x^{4} e^{\left (4 \, x e^{\left (-e^{2}\right )} - e^{2}\right )} + 16 \, x^{4} e^{\left (2 \, x e^{\left (-e^{2}\right )} - 2 \, e^{2}\right )} + 6 \, x^{3} e^{\left (4 \, x e^{\left (-e^{2}\right )}\right )} + 24 \, x^{3} e^{\left (2 \, x e^{\left (-e^{2}\right )} - e^{2}\right )} + 8 \, x^{3} e^{\left (-2 \, e^{2}\right )} + 12 \, x^{2} e^{\left (2 \, x e^{\left (-e^{2}\right )}\right )} + x^{2} e^{\left (4 \, x e^{\left (-e^{2}\right )} + e^{2}\right )} + 12 \, x^{2} e^{\left (-e^{2}\right )} + 2 \, x e^{\left (2 \, x e^{\left (-e^{2}\right )} + e^{2}\right )} + 6 \, x + e^{\left (e^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2)))/(25*x^3*exp(exp(2))*exp(2*x/exp(exp
(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp(exp(2)))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x, alg
orithm="giac")

[Out]

1/25*(8*x^6*e^(-2*e^2) + 12*x^5*e^(-e^2) + 6*x^4 + x^3*e^(e^2))/(8*x^5*e^(4*x*e^(-e^2) - 2*e^2) + 12*x^4*e^(4*
x*e^(-e^2) - e^2) + 16*x^4*e^(2*x*e^(-e^2) - 2*e^2) + 6*x^3*e^(4*x*e^(-e^2)) + 24*x^3*e^(2*x*e^(-e^2) - e^2) +
8*x^3*e^(-2*e^2) + 12*x^2*e^(2*x*e^(-e^2)) + x^2*e^(4*x*e^(-e^2) + e^2) + 12*x^2*e^(-e^2) + 2*x*e^(2*x*e^(-e^
2) + e^2) + 6*x + e^(e^2))

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maple [A]  time = 0.66, size = 21, normalized size = 0.88

method	result	size
norman	\(\frac {x^{3}}{25 \left (x \,{\mathrm e}^{2 x \,{\mathrm e}^{-{\mathrm e}^{2}}}+1\right )^{2}}\)	\(21\)
risch	\(\frac {x^{3}}{25 \left (x \,{\mathrm e}^{2 x \,{\mathrm e}^{-{\mathrm e}^{2}}}+1\right )^{2}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2)))/(25*x^3*exp(exp(2))*exp(2*x/exp(exp(2)))^
3+75*x^2*exp(exp(2))*exp(2*x/exp(exp(2)))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x,method=_RE
TURNVERBOSE)

[Out]

1/25*x^3/(x*exp(2*x/exp(exp(2)))+1)^2

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maxima [A]  time = 0.44, size = 34, normalized size = 1.42 \begin {gather*} \frac {x^{3}}{25 \, {\left (x^{2} e^{\left (4 \, x e^{\left (-e^{2}\right )}\right )} + 2 \, x e^{\left (2 \, x e^{\left (-e^{2}\right )}\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2)))/(25*x^3*exp(exp(2))*exp(2*x/exp(exp
(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp(exp(2)))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x, alg
orithm="maxima")

[Out]

1/25*x^3/(x^2*e^(4*x*e^(-e^2)) + 2*x*e^(2*x*e^(-e^2)) + 1)

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mupad [B]  time = 0.29, size = 34, normalized size = 1.42 \begin {gather*} \frac {x^3}{50\,x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-{\mathrm {e}}^2}}+25\,x^2\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{-{\mathrm {e}}^2}}+25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*exp(exp(2)) + exp(2*x*exp(-exp(2)))*(x^3*exp(exp(2)) - 4*x^4))/(25*exp(exp(2)) + 75*x*exp(2*x*exp(-
exp(2)))*exp(exp(2)) + 75*x^2*exp(4*x*exp(-exp(2)))*exp(exp(2)) + 25*x^3*exp(6*x*exp(-exp(2)))*exp(exp(2))),x)

[Out]

x^3/(50*x*exp(2*x*exp(-exp(2))) + 25*x^2*exp(4*x*exp(-exp(2))) + 25)

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sympy [A]  time = 0.15, size = 32, normalized size = 1.33 \begin {gather*} \frac {x^{3}}{25 x^{2} e^{\frac {4 x}{e^{e^{2}}}} + 50 x e^{\frac {2 x}{e^{e^{2}}}} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3*exp(exp(2))-4*x**4)*exp(2*x/exp(exp(2)))+3*x**2*exp(exp(2)))/(25*x**3*exp(exp(2))*exp(2*x/exp
(exp(2)))**3+75*x**2*exp(exp(2))*exp(2*x/exp(exp(2)))**2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2)))
,x)

[Out]

x**3/(25*x**2*exp(4*x*exp(-exp(2))) + 50*x*exp(2*x*exp(-exp(2))) + 25)

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