3.87.76 \(\int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}} (-75+4 x+50 x^2-2 x^3)}{625 x^4-50 x^5+x^6} \, dx\)

Optimal. Leaf size=21 \[ e^{\frac {e^{x^2}}{25 x^3-x^4}} \]

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Rubi [F]  time = 1.71, 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^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}} \left (-75+4 x+50 x^2-2 x^3\right )}{625 x^4-50 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x^2 - E^x^2/(-25*x^3 + x^4))*(-75 + 4*x + 50*x^2 - 2*x^3))/(625*x^4 - 50*x^5 + x^6),x]

[Out]

Defer[Int][E^(x^2 - E^x^2/(-25*x^3 + x^4))/(-25 + x)^2, x]/15625 - (2*Defer[Int][E^(x^2 - E^x^2/(-25*x^3 + x^4
))/(-25 + x), x])/625 - (3*Defer[Int][E^(x^2 - E^x^2/(-25*x^3 + x^4))/x^4, x])/25 - (2*Defer[Int][E^(x^2 - E^x
^2/(-25*x^3 + x^4))/x^3, x])/625 + (1249*Defer[Int][E^(x^2 - E^x^2/(-25*x^3 + x^4))/x^2, x])/15625 + (2*Defer[
Int][E^(x^2 - E^x^2/(-25*x^3 + x^4))/x, x])/625

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}} \left (-75+4 x+50 x^2-2 x^3\right )}{x^4 \left (625-50 x+x^2\right )} \, dx\\ &=\int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}} \left (-75+4 x+50 x^2-2 x^3\right )}{(-25+x)^2 x^4} \, dx\\ &=\int \left (\frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{15625 (-25+x)^2}-\frac {2 e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{625 (-25+x)}-\frac {3 e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{25 x^4}-\frac {2 e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{625 x^3}+\frac {1249 e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{15625 x^2}+\frac {2 e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{625 x}\right ) \, dx\\ &=\frac {\int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{(-25+x)^2} \, dx}{15625}-\frac {2}{625} \int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{-25+x} \, dx-\frac {2}{625} \int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{x^3} \, dx+\frac {2}{625} \int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{x} \, dx+\frac {1249 \int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{x^2} \, dx}{15625}-\frac {3}{25} \int \frac {e^{x^2-\frac {e^{x^2}}{-25 x^3+x^4}}}{x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 17, normalized size = 0.81 \begin {gather*} e^{-\frac {e^{x^2}}{(-25+x) x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2 - E^x^2/(-25*x^3 + x^4))*(-75 + 4*x + 50*x^2 - 2*x^3))/(625*x^4 - 50*x^5 + x^6),x]

[Out]

E^(-(E^x^2/((-25 + x)*x^3)))

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fricas [A]  time = 1.11, size = 34, normalized size = 1.62 \begin {gather*} e^{\left (-x^{2} + \frac {x^{6} - 25 \, x^{5} - e^{\left (x^{2}\right )}}{x^{4} - 25 \, x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+50*x^2+4*x-75)*exp(x^2)*exp(-exp(x^2)/(x^4-25*x^3))/(x^6-50*x^5+625*x^4),x, algorithm="frica
s")

[Out]

e^(-x^2 + (x^6 - 25*x^5 - e^(x^2))/(x^4 - 25*x^3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, x^{3} - 50 \, x^{2} - 4 \, x + 75\right )} e^{\left (x^{2} - \frac {e^{\left (x^{2}\right )}}{x^{4} - 25 \, x^{3}}\right )}}{x^{6} - 50 \, x^{5} + 625 \, x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+50*x^2+4*x-75)*exp(x^2)*exp(-exp(x^2)/(x^4-25*x^3))/(x^6-50*x^5+625*x^4),x, algorithm="giac"
)

[Out]

integrate(-(2*x^3 - 50*x^2 - 4*x + 75)*e^(x^2 - e^(x^2)/(x^4 - 25*x^3))/(x^6 - 50*x^5 + 625*x^4), x)

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maple [A]  time = 0.09, size = 16, normalized size = 0.76

method	result	size
risch	\({\mathrm e}^{-\frac {{\mathrm e}^{x^{2}}}{x^{3} \left (x -25\right )}}\)	\(16\)
norman	\(\frac {x^{4} {\mathrm e}^{-\frac {{\mathrm e}^{x^{2}}}{x^{4}-25 x^{3}}}-25 x^{3} {\mathrm e}^{-\frac {{\mathrm e}^{x^{2}}}{x^{4}-25 x^{3}}}}{x^{3} \left (x -25\right )}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3+50*x^2+4*x-75)*exp(x^2)*exp(-exp(x^2)/(x^4-25*x^3))/(x^6-50*x^5+625*x^4),x,method=_RETURNVERBOSE)

[Out]

exp(-exp(x^2)/x^3/(x-25))

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maxima [B]  time = 0.49, size = 40, normalized size = 1.90 \begin {gather*} e^{\left (-\frac {e^{\left (x^{2}\right )}}{15625 \, {\left (x - 25\right )}} + \frac {e^{\left (x^{2}\right )}}{15625 \, x} + \frac {e^{\left (x^{2}\right )}}{625 \, x^{2}} + \frac {e^{\left (x^{2}\right )}}{25 \, x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+50*x^2+4*x-75)*exp(x^2)*exp(-exp(x^2)/(x^4-25*x^3))/(x^6-50*x^5+625*x^4),x, algorithm="maxim
a")

[Out]

e^(-1/15625*e^(x^2)/(x - 25) + 1/15625*e^(x^2)/x + 1/625*e^(x^2)/x^2 + 1/25*e^(x^2)/x^3)

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mupad [B]  time = 5.87, size = 19, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}}{25\,x^3-x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x^2)/(25*x^3 - x^4))*exp(x^2)*(4*x + 50*x^2 - 2*x^3 - 75))/(625*x^4 - 50*x^5 + x^6),x)

[Out]

exp(exp(x^2)/(25*x^3 - x^4))

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sympy [A]  time = 0.42, size = 15, normalized size = 0.71 \begin {gather*} e^{- \frac {e^{x^{2}}}{x^{4} - 25 x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3+50*x**2+4*x-75)*exp(x**2)*exp(-exp(x**2)/(x**4-25*x**3))/(x**6-50*x**5+625*x**4),x)

[Out]

exp(-exp(x**2)/(x**4 - 25*x**3))

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