3.84.91 \(\int \frac {e^{\frac {5 x-9 x \log (x)}{e^x (-5+x)+9 e^x \log (x)}} (-25+16 x-5 x^2+(90-90 x+9 x^2) \log (x)+(-81+81 x) \log ^2(x))}{e^x (25-10 x+x^2)+e^x (-90+18 x) \log (x)+81 e^x \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ e^{\frac {e^{-x} x}{-1+\frac {x}{5-9 \log (x)}}} \]

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Rubi [F]  time = 10.48, 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 {\exp \left (\frac {5 x-9 x \log (x)}{e^x (-5+x)+9 e^x \log (x)}\right ) \left (-25+16 x-5 x^2+\left (90-90 x+9 x^2\right ) \log (x)+(-81+81 x) \log ^2(x)\right )}{e^x \left (25-10 x+x^2\right )+e^x (-90+18 x) \log (x)+81 e^x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((5*x - 9*x*Log[x])/(E^x*(-5 + x) + 9*E^x*Log[x]))*(-25 + 16*x - 5*x^2 + (90 - 90*x + 9*x^2)*Log[x] + (
-81 + 81*x)*Log[x]^2))/(E^x*(25 - 10*x + x^2) + E^x*(-90 + 18*x)*Log[x] + 81*E^x*Log[x]^2),x]

[Out]

-Defer[Int][E^(-x + (5*x - 9*x*Log[x])/(E^x*(-5 + x + 9*Log[x]))), x] + Defer[Int][E^(-x + (5*x - 9*x*Log[x])/
(E^x*(-5 + x + 9*Log[x])))*x, x] - 9*Defer[Int][(E^(-x + (5*x - 9*x*Log[x])/(E^x*(-5 + x + 9*Log[x])))*x)/(-5
+ x + 9*Log[x])^2, x] - Defer[Int][(E^(-x + (5*x - 9*x*Log[x])/(E^x*(-5 + x + 9*Log[x])))*x^2)/(-5 + x + 9*Log
[x])^2, x] + 2*Defer[Int][(E^(-x + (5*x - 9*x*Log[x])/(E^x*(-5 + x + 9*Log[x])))*x)/(-5 + x + 9*Log[x]), x] -
Defer[Int][(E^(-x + (5*x - 9*x*Log[x])/(E^x*(-5 + x + 9*Log[x])))*x^2)/(-5 + x + 9*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) \left (-25+16 x-5 x^2+\left (90-90 x+9 x^2\right ) \log (x)+(-81+81 x) \log ^2(x)\right )}{(5-x-9 \log (x))^2} \, dx\\ &=\int \left (-\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right )+\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x-\frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x (9+x)}{(-5+x+9 \log (x))^2}-\frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) (-2+x) x}{-5+x+9 \log (x)}\right ) \, dx\\ &=-\int \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) \, dx+\int \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x \, dx-\int \frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x (9+x)}{(-5+x+9 \log (x))^2} \, dx-\int \frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) (-2+x) x}{-5+x+9 \log (x)} \, dx\\ &=-\int \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) \, dx+\int \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x \, dx-\int \left (\frac {9 \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x}{(-5+x+9 \log (x))^2}+\frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x^2}{(-5+x+9 \log (x))^2}\right ) \, dx-\int \left (-\frac {2 \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x}{-5+x+9 \log (x)}+\frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x^2}{-5+x+9 \log (x)}\right ) \, dx\\ &=2 \int \frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x}{-5+x+9 \log (x)} \, dx-9 \int \frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x}{(-5+x+9 \log (x))^2} \, dx-\int \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) \, dx+\int \exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x \, dx-\int \frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x^2}{(-5+x+9 \log (x))^2} \, dx-\int \frac {\exp \left (-x+\frac {e^{-x} (5 x-9 x \log (x))}{-5+x+9 \log (x)}\right ) x^2}{-5+x+9 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 39, normalized size = 1.70 \begin {gather*} e^{\frac {5 e^{-x} x}{-5+x+9 \log (x)}} x^{-\frac {9 e^{-x} x}{-5+x+9 \log (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((5*x - 9*x*Log[x])/(E^x*(-5 + x) + 9*E^x*Log[x]))*(-25 + 16*x - 5*x^2 + (90 - 90*x + 9*x^2)*Log[
x] + (-81 + 81*x)*Log[x]^2))/(E^x*(25 - 10*x + x^2) + E^x*(-90 + 18*x)*Log[x] + 81*E^x*Log[x]^2),x]

[Out]

E^((5*x)/(E^x*(-5 + x + 9*Log[x])))/x^((9*x)/(E^x*(-5 + x + 9*Log[x])))

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fricas [A]  time = 1.02, size = 27, normalized size = 1.17 \begin {gather*} e^{\left (-\frac {9 \, x \log \relax (x) - 5 \, x}{{\left (x - 5\right )} e^{x} + 9 \, e^{x} \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x-81)*log(x)^2+(9*x^2-90*x+90)*log(x)-5*x^2+16*x-25)*exp((-9*x*log(x)+5*x)/(9*exp(x)*log(x)+(x-
5)*exp(x)))/(81*exp(x)*log(x)^2+(18*x-90)*exp(x)*log(x)+(x^2-10*x+25)*exp(x)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.31, size = 44, normalized size = 1.91 \begin {gather*} e^{\left (-\frac {9 \, x \log \relax (x)}{x e^{x} + 9 \, e^{x} \log \relax (x) - 5 \, e^{x}} + \frac {5 \, x}{x e^{x} + 9 \, e^{x} \log \relax (x) - 5 \, e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x-81)*log(x)^2+(9*x^2-90*x+90)*log(x)-5*x^2+16*x-25)*exp((-9*x*log(x)+5*x)/(9*exp(x)*log(x)+(x-
5)*exp(x)))/(81*exp(x)*log(x)^2+(18*x-90)*exp(x)*log(x)+(x^2-10*x+25)*exp(x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 24, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{-\frac {x \left (9 \ln \relax (x )-5\right ) {\mathrm e}^{-x}}{9 \ln \relax (x )+x -5}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((81*x-81)*ln(x)^2+(9*x^2-90*x+90)*ln(x)-5*x^2+16*x-25)*exp((-9*x*ln(x)+5*x)/(9*exp(x)*ln(x)+(x-5)*exp(x))
)/(81*exp(x)*ln(x)^2+(18*x-90)*exp(x)*ln(x)+(x^2-10*x+25)*exp(x)),x,method=_RETURNVERBOSE)

[Out]

exp(-x*(9*ln(x)-5)*exp(-x)/(9*ln(x)+x-5))

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maxima [B]  time = 0.60, size = 73, normalized size = 3.17 \begin {gather*} e^{\left (-9 \, e^{\left (-x\right )} \log \relax (x) + \frac {81 \, \log \relax (x)^{2}}{{\left (x - 5\right )} e^{x} + 9 \, e^{x} \log \relax (x)} - \frac {90 \, \log \relax (x)}{{\left (x - 5\right )} e^{x} + 9 \, e^{x} \log \relax (x)} + \frac {25}{{\left (x - 5\right )} e^{x} + 9 \, e^{x} \log \relax (x)} + 5 \, e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x-81)*log(x)^2+(9*x^2-90*x+90)*log(x)-5*x^2+16*x-25)*exp((-9*x*log(x)+5*x)/(9*exp(x)*log(x)+(x-
5)*exp(x)))/(81*exp(x)*log(x)^2+(18*x-90)*exp(x)*log(x)+(x^2-10*x+25)*exp(x)),x, algorithm="maxima")

[Out]

e^(-9*e^(-x)*log(x) + 81*log(x)^2/((x - 5)*e^x + 9*e^x*log(x)) - 90*log(x)/((x - 5)*e^x + 9*e^x*log(x)) + 25/(
(x - 5)*e^x + 9*e^x*log(x)) + 5*e^(-x))

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mupad [B]  time = 5.54, size = 45, normalized size = 1.96 \begin {gather*} {\mathrm {e}}^{\frac {5\,x}{9\,{\mathrm {e}}^x\,\ln \relax (x)-5\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {9\,x\,\ln \relax (x)}{9\,{\mathrm {e}}^x\,\ln \relax (x)-5\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((5*x - 9*x*log(x))/(exp(x)*(x - 5) + 9*exp(x)*log(x)))*(16*x + log(x)*(9*x^2 - 90*x + 90) - 5*x^2 + l
og(x)^2*(81*x - 81) - 25))/(exp(x)*(x^2 - 10*x + 25) + 81*exp(x)*log(x)^2 + exp(x)*log(x)*(18*x - 90)),x)

[Out]

exp((5*x)/(9*exp(x)*log(x) - 5*exp(x) + x*exp(x)))*exp(-(9*x*log(x))/(9*exp(x)*log(x) - 5*exp(x) + x*exp(x)))

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sympy [A]  time = 1.39, size = 26, normalized size = 1.13 \begin {gather*} e^{\frac {- 9 x \log {\relax (x )} + 5 x}{\left (x - 5\right ) e^{x} + 9 e^{x} \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x-81)*ln(x)**2+(9*x**2-90*x+90)*ln(x)-5*x**2+16*x-25)*exp((-9*x*ln(x)+5*x)/(9*exp(x)*ln(x)+(x-5
)*exp(x)))/(81*exp(x)*ln(x)**2+(18*x-90)*exp(x)*ln(x)+(x**2-10*x+25)*exp(x)),x)

[Out]

exp((-9*x*log(x) + 5*x)/((x - 5)*exp(x) + 9*exp(x)*log(x)))

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