3.21.67 \(\int \frac {e^{5 e^{-\frac {-2-3 x+x^2}{-3+x}}} (-275+150 x-25 x^2)}{e^{5 e^{-\frac {-2-3 x+x^2}{-3+x}}+\frac {-2-3 x+x^2}{-3+x}} (9-6 x+x^2)+e^{\frac {-2-3 x+x^2}{-3+x}} (63-42 x+7 x^2)} \, dx\) [2067]

Optimal. Leaf size=26 \[ 5 \log \left (-7-e^{5 e^{-\frac {2}{3-x}-x}}\right ) \]

[Out]

5*ln(-7-exp(5/exp(x+2/(3-x))))

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Rubi [F]
time = 8.47, 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^{-\frac {-2-3 x+x^2}{-3+x}}} \left (-275+150 x-25 x^2\right )}{\exp \left (5 e^{-\frac {-2-3 x+x^2}{-3+x}}+\frac {-2-3 x+x^2}{-3+x}\right ) \left (9-6 x+x^2\right )+e^{\frac {-2-3 x+x^2}{-3+x}} \left (63-42 x+7 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5/E^((-2 - 3*x + x^2)/(-3 + x)))*(-275 + 150*x - 25*x^2))/(E^(5/E^((-2 - 3*x + x^2)/(-3 + x)) + (-2 -
3*x + x^2)/(-3 + x))*(9 - 6*x + x^2) + E^((-2 - 3*x + x^2)/(-3 + x))*(63 - 42*x + 7*x^2)),x]

[Out]

-25*Defer[Int][E^((2 + 3*x - x^2)/(-3 + x)), x] + 175*Defer[Int][E^((2 + 3*x - x^2)/(-3 + x))/(7 + E^(5*E^((2
+ 3*x - x^2)/(-3 + x)))), x] - 50*Defer[Int][E^((2 + 3*x - x^2)/(-3 + x))/(-3 + x)^2, x] + 350*Defer[Int][E^((
2 + 3*x - x^2)/(-3 + x))/((7 + E^(5*E^(2/(-3 + x) + (3*x)/(-3 + x) - x^2/(-3 + x))))*(-3 + x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 e^{\frac {2+3 x-x^2}{-3+x}} \left (-11+6 x-x^2\right )}{\left (1+7 e^{-5 e^{\frac {2+3 x-x^2}{-3+x}}}\right ) (3-x)^2} \, dx\\ &=25 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}} \left (-11+6 x-x^2\right )}{\left (1+7 e^{-5 e^{\frac {2+3 x-x^2}{-3+x}}}\right ) (3-x)^2} \, dx\\ &=25 \int \left (\frac {e^{\frac {2+3 x-x^2}{-3+x}} \left (-11+6 x-x^2\right )}{(-3+x)^2}+\frac {7 e^{\frac {2+3 x-x^2}{-3+x}} \left (11-6 x+x^2\right )}{\left (7+e^{5 e^{\frac {2}{-3+x}+\frac {3 x}{-3+x}-\frac {x^2}{-3+x}}}\right ) (-3+x)^2}\right ) \, dx\\ &=25 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}} \left (-11+6 x-x^2\right )}{(-3+x)^2} \, dx+175 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}} \left (11-6 x+x^2\right )}{\left (7+e^{5 e^{\frac {2}{-3+x}+\frac {3 x}{-3+x}-\frac {x^2}{-3+x}}}\right ) (-3+x)^2} \, dx\\ &=25 \int \left (-e^{\frac {2+3 x-x^2}{-3+x}}-\frac {2 e^{\frac {2+3 x-x^2}{-3+x}}}{(-3+x)^2}\right ) \, dx+175 \int \left (\frac {e^{\frac {2+3 x-x^2}{-3+x}}}{7+e^{5 e^{\frac {2}{-3+x}+\frac {3 x}{-3+x}-\frac {x^2}{-3+x}}}}+\frac {2 e^{\frac {2+3 x-x^2}{-3+x}}}{\left (7+e^{5 e^{\frac {2}{-3+x}+\frac {3 x}{-3+x}-\frac {x^2}{-3+x}}}\right ) (-3+x)^2}\right ) \, dx\\ &=-\left (25 \int e^{\frac {2+3 x-x^2}{-3+x}} \, dx\right )-50 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}}}{(-3+x)^2} \, dx+175 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}}}{7+e^{5 e^{\frac {2}{-3+x}+\frac {3 x}{-3+x}-\frac {x^2}{-3+x}}}} \, dx+350 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}}}{\left (7+e^{5 e^{\frac {2}{-3+x}+\frac {3 x}{-3+x}-\frac {x^2}{-3+x}}}\right ) (-3+x)^2} \, dx\\ &=-\left (25 \int e^{\frac {2+3 x-x^2}{-3+x}} \, dx\right )-50 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}}}{(-3+x)^2} \, dx+175 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}}}{7+e^{5 e^{\frac {2+3 x-x^2}{-3+x}}}} \, dx+350 \int \frac {e^{\frac {2+3 x-x^2}{-3+x}}}{\left (7+e^{5 e^{\frac {2}{-3+x}+\frac {3 x}{-3+x}-\frac {x^2}{-3+x}}}\right ) (-3+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 1.01, size = 22, normalized size = 0.85 \begin {gather*} 5 \log \left (7+e^{5 e^{\frac {2}{-3+x}-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5/E^((-2 - 3*x + x^2)/(-3 + x)))*(-275 + 150*x - 25*x^2))/(E^(5/E^((-2 - 3*x + x^2)/(-3 + x)) +
(-2 - 3*x + x^2)/(-3 + x))*(9 - 6*x + x^2) + E^((-2 - 3*x + x^2)/(-3 + x))*(63 - 42*x + 7*x^2)),x]

[Out]

5*Log[7 + E^(5*E^(2/(-3 + x) - x))]

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Maple [A]
time = 0.10, size = 25, normalized size = 0.96

method	result	size
risch	\(5 \ln \left ({\mathrm e}^{5 \,{\mathrm e}^{-\frac {x^{2}-3 x -2}{x -3}}}+7\right )\)	\(25\)
norman	\(5 \ln \left ({\mathrm e}^{5 \,{\mathrm e}^{-\frac {x^{2}-3 x -2}{x -3}}}+7\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-25*x^2+150*x-275)*exp(5/exp((x^2-3*x-2)/(x-3)))/((x^2-6*x+9)*exp((x^2-3*x-2)/(x-3))*exp(5/exp((x^2-3*x-2
)/(x-3)))+(7*x^2-42*x+63)*exp((x^2-3*x-2)/(x-3))),x,method=_RETURNVERBOSE)

[Out]

5*ln(exp(5*exp(-(x^2-3*x-2)/(x-3)))+7)

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Maxima [A]
time = 0.71, size = 20, normalized size = 0.77 \begin {gather*} 5 \, \log \left (e^{\left (5 \, e^{\left (-x + \frac {2}{x - 3}\right )}\right )} + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x^2+150*x-275)*exp(5/exp((x^2-3*x-2)/(-3+x)))/((x^2-6*x+9)*exp((x^2-3*x-2)/(-3+x))*exp(5/exp((x
^2-3*x-2)/(-3+x)))+(7*x^2-42*x+63)*exp((x^2-3*x-2)/(-3+x))),x, algorithm="maxima")

[Out]

5*log(e^(5*e^(-x + 2/(x - 3))) + 7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (22) = 44\).
time = 0.39, size = 91, normalized size = 3.50 \begin {gather*} -\frac {5 \, {\left (x^{2} - {\left (x - 3\right )} \log \left (e^{\left (\frac {{\left ({\left (x^{2} - 3 \, x - 2\right )} e^{\left (\frac {x^{2} - 3 \, x - 2}{x - 3}\right )} + 5 \, x - 15\right )} e^{\left (-\frac {x^{2} - 3 \, x - 2}{x - 3}\right )}}{x - 3}\right )} + 7 \, e^{\left (\frac {x^{2} - 3 \, x - 2}{x - 3}\right )}\right ) - 3 \, x - 2\right )}}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x^2+150*x-275)*exp(5/exp((x^2-3*x-2)/(-3+x)))/((x^2-6*x+9)*exp((x^2-3*x-2)/(-3+x))*exp(5/exp((x
^2-3*x-2)/(-3+x)))+(7*x^2-42*x+63)*exp((x^2-3*x-2)/(-3+x))),x, algorithm="fricas")

[Out]

-5*(x^2 - (x - 3)*log(e^(((x^2 - 3*x - 2)*e^((x^2 - 3*x - 2)/(x - 3)) + 5*x - 15)*e^(-(x^2 - 3*x - 2)/(x - 3))
/(x - 3)) + 7*e^((x^2 - 3*x - 2)/(x - 3))) - 3*x - 2)/(x - 3)

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Sympy [A]
time = 0.25, size = 20, normalized size = 0.77 \begin {gather*} 5 \log {\left (e^{5 e^{- \frac {x^{2} - 3 x - 2}{x - 3}}} + 7 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x**2+150*x-275)*exp(5/exp((x**2-3*x-2)/(-3+x)))/((x**2-6*x+9)*exp((x**2-3*x-2)/(-3+x))*exp(5/ex
p((x**2-3*x-2)/(-3+x)))+(7*x**2-42*x+63)*exp((x**2-3*x-2)/(-3+x))),x)

[Out]

5*log(exp(5*exp(-(x**2 - 3*x - 2)/(x - 3))) + 7)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (22) = 44\).
time = 1.03, size = 114, normalized size = 4.38 \begin {gather*} \frac {5 \, {\left (5 \, x^{2} e^{\left (-\frac {x^{2} - 3 \, x - 2}{x - 3}\right )} - 10 \, x e^{\left (-\frac {x^{2} - 3 \, x - 2}{x - 3}\right )} + x \log \left (e^{\left (5 \, e^{\left (-\frac {x^{2} - 3 \, x - 2}{x - 3}\right )}\right )} + 7\right ) - 25 \, e^{\left (-\frac {x^{2} - 3 \, x - 2}{x - 3}\right )} - 3 \, \log \left (e^{\left (5 \, e^{\left (-\frac {x^{2} - 3 \, x - 2}{x - 3}\right )}\right )} + 7\right )\right )}}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*x^2+150*x-275)*exp(5/exp((x^2-3*x-2)/(-3+x)))/((x^2-6*x+9)*exp((x^2-3*x-2)/(-3+x))*exp(5/exp((x
^2-3*x-2)/(-3+x)))+(7*x^2-42*x+63)*exp((x^2-3*x-2)/(-3+x))),x, algorithm="giac")

[Out]

5*(5*x^2*e^(-(x^2 - 3*x - 2)/(x - 3)) - 10*x*e^(-(x^2 - 3*x - 2)/(x - 3)) + x*log(e^(5*e^(-(x^2 - 3*x - 2)/(x
- 3))) + 7) - 25*e^(-(x^2 - 3*x - 2)/(x - 3)) - 3*log(e^(5*e^(-(x^2 - 3*x - 2)/(x - 3))) + 7))/(x - 3)

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Mupad [B]
time = 1.69, size = 36, normalized size = 1.38 \begin {gather*} \ln \left ({\left ({\mathrm {e}}^{5\,{\mathrm {e}}^{\frac {3\,x}{x-3}}\,{\mathrm {e}}^{-\frac {x^2}{x-3}}\,{\mathrm {e}}^{\frac {2}{x-3}}}+7\right )}^5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5*exp((3*x - x^2 + 2)/(x - 3)))*(25*x^2 - 150*x + 275))/(exp(-(3*x - x^2 + 2)/(x - 3))*(7*x^2 - 42*x
+ 63) + exp(5*exp((3*x - x^2 + 2)/(x - 3)))*exp(-(3*x - x^2 + 2)/(x - 3))*(x^2 - 6*x + 9)),x)

[Out]

log((exp(5*exp((3*x)/(x - 3))*exp(-x^2/(x - 3))*exp(2/(x - 3))) + 7)^5)

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