3.95.34 \(\int \frac {-8 e+e^{5-x} (-27+36 x)}{-24 e+27 e^{5-x} x} \, dx\)

Optimal. Leaf size=25 \[ \frac {x}{3}-\log \left (-\frac {4}{9}+\frac {1}{2} e^{4-x} x\right ) \]

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Rubi [F]  time = 0.25, 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 {-8 e+e^{5-x} (-27+36 x)}{-24 e+27 e^{5-x} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*E + E^(5 - x)*(-27 + 36*x))/(-24*E + 27*E^(5 - x)*x),x]

[Out]

x/3 + 9*E^4*Defer[Int][(8*E^x - 9*E^4*x)^(-1), x] + 9*E^4*Defer[Int][x/(-8*E^x + 9*E^4*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{3}+\frac {9 e^4 (-1+x)}{-8 e^x+9 e^4 x}\right ) \, dx\\ &=\frac {x}{3}+\left (9 e^4\right ) \int \frac {-1+x}{-8 e^x+9 e^4 x} \, dx\\ &=\frac {x}{3}+\left (9 e^4\right ) \int \left (\frac {1}{8 e^x-9 e^4 x}+\frac {x}{-8 e^x+9 e^4 x}\right ) \, dx\\ &=\frac {x}{3}+\left (9 e^4\right ) \int \frac {1}{8 e^x-9 e^4 x} \, dx+\left (9 e^4\right ) \int \frac {x}{-8 e^x+9 e^4 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{3} \left (4 x-3 \log \left (8 e^x-9 e^4 x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*E + E^(5 - x)*(-27 + 36*x))/(-24*E + 27*E^(5 - x)*x),x]

[Out]

(4*x - 3*Log[8*E^x - 9*E^4*x])/3

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fricas [A]  time = 0.56, size = 29, normalized size = 1.16 \begin {gather*} \frac {1}{3} \, x - \log \relax (x) - \log \left (\frac {9 \, x e^{\left (-x + 5\right )} - 8 \, e}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x-27)*exp(5-x)-8*exp(1))/(27*x*exp(5-x)-24*exp(1)),x, algorithm="fricas")

[Out]

1/3*x - log(x) - log((9*x*e^(-x + 5) - 8*e)/x)

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giac [A]  time = 0.24, size = 18, normalized size = 0.72 \begin {gather*} \frac {1}{3} \, x - \log \left (9 \, x e^{\left (-x + 4\right )} - 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x-27)*exp(5-x)-8*exp(1))/(27*x*exp(5-x)-24*exp(1)),x, algorithm="giac")

[Out]

1/3*x - log(9*x*e^(-x + 4) - 8)

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maple [A]  time = 0.15, size = 22, normalized size = 0.88

method	result	size
norman	\(\frac {x}{3}-\ln \left (-27 x \,{\mathrm e}^{5-x}+24 \,{\mathrm e}\right )\)	\(22\)
risch	\(\frac {x}{3}-\ln \relax (x )+5-\ln \left ({\mathrm e}^{5-x}-\frac {8 \,{\mathrm e}}{9 x}\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((36*x-27)*exp(5-x)-8*exp(1))/(27*x*exp(5-x)-24*exp(1)),x,method=_RETURNVERBOSE)

[Out]

1/3*x-ln(-27*x*exp(5-x)+24*exp(1))

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maxima [A]  time = 0.40, size = 15, normalized size = 0.60 \begin {gather*} \frac {4}{3} \, x - \log \left (-\frac {9}{8} \, x e^{4} + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x-27)*exp(5-x)-8*exp(1))/(27*x*exp(5-x)-24*exp(1)),x, algorithm="maxima")

[Out]

4/3*x - log(-9/8*x*e^4 + e^x)

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mupad [B]  time = 5.93, size = 21, normalized size = 0.84 \begin {gather*} \frac {x}{3}-\ln \left (9\,x\,{\mathrm {e}}^{5-x}-8\,\mathrm {e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*exp(1) - exp(5 - x)*(36*x - 27))/(24*exp(1) - 27*x*exp(5 - x)),x)

[Out]

x/3 - log(9*x*exp(5 - x) - 8*exp(1))

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sympy [A]  time = 0.19, size = 20, normalized size = 0.80 \begin {gather*} \frac {x}{3} - \log {\relax (x )} - \log {\left (e^{5 - x} - \frac {8 e}{9 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x-27)*exp(5-x)-8*exp(1))/(27*x*exp(5-x)-24*exp(1)),x)

[Out]

x/3 - log(x) - log(exp(5 - x) - 8*E/(9*x))

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