3.72.89 \(\int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+(-8 x+4 x^2) \log (x)+x \log ^2(x)}{-3+x}} (-48+76 x-48 x^2+8 x^3+(18-22 x+4 x^2) \log (x)-3 \log ^2(x))}{9-6 x+x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{-x+\frac {x \left (5+x+(4-2 x-\log (x))^2\right )}{-3+x}} \]

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Rubi [F]  time = 5.44, 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 (-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}\right ) \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x + (21*x - 15*x^2 + 4*x^3 + (-8*x + 4*x^2)*Log[x] + x*Log[x]^2)/(-3 + x))*(-48 + 76*x - 48*x^2 + 8*x
^3 + (18 - 22*x + 4*x^2)*Log[x] - 3*Log[x]^2))/(9 - 6*x + x^2),x]

[Out]

-36*Defer[Int][E^((x*(24 - 16*x + 4*x^2 - 8*Log[x] + 4*x*Log[x] + Log[x]^2))/(-3 + x))/(-3 + x)^2, x] + 4*Defe
r[Int][E^((x*(24 - 16*x + 4*x^2 - 8*Log[x] + 4*x*Log[x] + Log[x]^2))/(-3 + x))/(-3 + x), x] + 8*Defer[Int][E^(
(x*(24 - 16*x + 4*x^2 - 8*Log[x] + 4*x*Log[x] + Log[x]^2))/(-3 + x))*x, x] + 4*Defer[Int][E^((x*(24 - 16*x + 4
*x^2 - 8*Log[x] + 4*x*Log[x] + Log[x]^2))/(-3 + x))*Log[x], x] - 12*Defer[Int][(E^((x*(24 - 16*x + 4*x^2 - 8*L
og[x] + 4*x*Log[x] + Log[x]^2))/(-3 + x))*Log[x])/(-3 + x)^2, x] + 2*Defer[Int][(E^((x*(24 - 16*x + 4*x^2 - 8*
Log[x] + 4*x*Log[x] + Log[x]^2))/(-3 + x))*Log[x])/(-3 + x), x] - 3*Defer[Int][(E^((x*(24 - 16*x + 4*x^2 - 8*L
og[x] + 4*x*Log[x] + Log[x]^2))/(-3 + x))*Log[x]^2)/(-3 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}\right ) \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{(-3+x)^2} \, dx\\ &=\int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{(3-x)^2} \, dx\\ &=\int \left (-\frac {48 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {76 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x}{(-3+x)^2}-\frac {48 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^2}{(-3+x)^2}+\frac {8 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^3}{(-3+x)^2}+\frac {2 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) (-1+x) (-9+2 x) \log (x)}{(-3+x)^2}-\frac {3 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) (-1+x) (-9+2 x) \log (x)}{(-3+x)^2} \, dx-3 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2} \, dx+8 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^3}{(-3+x)^2} \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x^2}{(-3+x)^2} \, dx+76 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x}{(-3+x)^2} \, dx\\ &=2 \int \left (2 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)-\frac {6 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{(-3+x)^2}+\frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{-3+x}\right ) \, dx-3 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2} \, dx+8 \int \left (6 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )+\frac {27 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {27 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x}+\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x\right ) \, dx-48 \int \left (\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )+\frac {9 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {6 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x}\right ) \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx+76 \int \left (\frac {3 \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2}+\frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{-3+x} \, dx-3 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log ^2(x)}{(-3+x)^2} \, dx+4 \int \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x) \, dx+8 \int \exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) x \, dx-12 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right ) \log (x)}{(-3+x)^2} \, dx-48 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx+76 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x} \, dx+216 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx+216 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x} \, dx+228 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx-288 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{-3+x} \, dx-432 \int \frac {\exp \left (\frac {x \left (24-16 x+4 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right )}{-3+x}\right )}{(-3+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.99, size = 38, normalized size = 1.41 \begin {gather*} e^{\frac {x \left (4 \left (6-4 x+x^2\right )+\log ^2(x)\right )}{-3+x}} x^{\frac {4 (-2+x) x}{-3+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x + (21*x - 15*x^2 + 4*x^3 + (-8*x + 4*x^2)*Log[x] + x*Log[x]^2)/(-3 + x))*(-48 + 76*x - 48*x^2
+ 8*x^3 + (18 - 22*x + 4*x^2)*Log[x] - 3*Log[x]^2))/(9 - 6*x + x^2),x]

[Out]

E^((x*(4*(6 - 4*x + x^2) + Log[x]^2))/(-3 + x))*x^((4*(-2 + x)*x)/(-3 + x))

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fricas [A]  time = 0.54, size = 38, normalized size = 1.41 \begin {gather*} e^{\left (\frac {4 \, x^{3} + x \log \relax (x)^{2} - 16 \, x^{2} + 4 \, {\left (x^{2} - 2 \, x\right )} \log \relax (x) + 24 \, x}{x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x*log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-1
5*x^2+21*x)/(x-3))/(x^2-6*x+9)/exp(x),x, algorithm="fricas")

[Out]

e^((4*x^3 + x*log(x)^2 - 16*x^2 + 4*(x^2 - 2*x)*log(x) + 24*x)/(x - 3))

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giac [A]  time = 0.33, size = 39, normalized size = 1.44 \begin {gather*} e^{\left (\frac {4 \, x^{3} + 4 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} - 16 \, x^{2} - 8 \, x \log \relax (x) + 24 \, x}{x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x*log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-1
5*x^2+21*x)/(x-3))/(x^2-6*x+9)/exp(x),x, algorithm="giac")

[Out]

e^((4*x^3 + 4*x^2*log(x) + x*log(x)^2 - 16*x^2 - 8*x*log(x) + 24*x)/(x - 3))

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maple [A]  time = 0.16, size = 32, normalized size = 1.19

method	result	size
risch	\({\mathrm e}^{\frac {x \left (\ln \relax (x )^{2}+4 x \ln \relax (x )+4 x^{2}-8 \ln \relax (x )-16 x +24\right )}{x -3}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*ln(x)^2+(4*x^2-22*x+18)*ln(x)+8*x^3-48*x^2+76*x-48)*exp((x*ln(x)^2+(4*x^2-8*x)*ln(x)+4*x^3-15*x^2+21*x
)/(x-3))/(x^2-6*x+9)/exp(x),x,method=_RETURNVERBOSE)

[Out]

exp(x*(ln(x)^2+4*x*ln(x)+4*x^2-8*ln(x)-16*x+24)/(x-3))

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maxima [B]  time = 0.51, size = 51, normalized size = 1.89 \begin {gather*} x^{4} e^{\left (4 \, x^{2} + 4 \, x \log \relax (x) + \log \relax (x)^{2} - 4 \, x + \frac {3 \, \log \relax (x)^{2}}{x - 3} + \frac {12 \, \log \relax (x)}{x - 3} + \frac {36}{x - 3} + 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x*log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-1
5*x^2+21*x)/(x-3))/(x^2-6*x+9)/exp(x),x, algorithm="maxima")

[Out]

x^4*e^(4*x^2 + 4*x*log(x) + log(x)^2 - 4*x + 3*log(x)^2/(x - 3) + 12*log(x)/(x - 3) + 36/(x - 3) + 12)

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mupad [B]  time = 4.62, size = 68, normalized size = 2.52 \begin {gather*} \frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {21\,x}{x-3}}\,{\mathrm {e}}^{\frac {x\,{\ln \relax (x)}^2}{x-3}}\,{\mathrm {e}}^{\frac {4\,x^3}{x-3}}\,{\mathrm {e}}^{-\frac {15\,x^2}{x-3}}}{x^{\frac {4\,\left (2\,x-x^2\right )}{x-3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*exp((21*x + x*log(x)^2 - log(x)*(8*x - 4*x^2) - 15*x^2 + 4*x^3)/(x - 3))*(76*x - 3*log(x)^2 + log
(x)*(4*x^2 - 22*x + 18) - 48*x^2 + 8*x^3 - 48))/(x^2 - 6*x + 9),x)

[Out]

(exp(-x)*exp((21*x)/(x - 3))*exp((x*log(x)^2)/(x - 3))*exp((4*x^3)/(x - 3))*exp(-(15*x^2)/(x - 3)))/x^((4*(2*x
- x^2))/(x - 3))

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sympy [A]  time = 61.18, size = 39, normalized size = 1.44 \begin {gather*} e^{- x} e^{\frac {4 x^{3} - 15 x^{2} + x \log {\relax (x )}^{2} + 21 x + \left (4 x^{2} - 8 x\right ) \log {\relax (x )}}{x - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*ln(x)**2+(4*x**2-22*x+18)*ln(x)+8*x**3-48*x**2+76*x-48)*exp((x*ln(x)**2+(4*x**2-8*x)*ln(x)+4*x**
3-15*x**2+21*x)/(x-3))/(x**2-6*x+9)/exp(x),x)

[Out]

exp(-x)*exp((4*x**3 - 15*x**2 + x*log(x)**2 + 21*x + (4*x**2 - 8*x)*log(x))/(x - 3))

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