3.74.13 \(\int \frac {e^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x))}{16 x^5+8 x^6+x^7+(8 x^5+2 x^6) \log ^2(x)+x^5 \log ^4(x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {2 e^{5+\frac {x}{\frac {4+x}{\log (x)}+\log (x)}}}{x^4} \]

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Rubi [F]  time = 2.65, 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^{\frac {20+5 x+x \log (x)+5 \log ^2(x)}{4+x+\log ^2(x)}} \left (-128-56 x-6 x^2+8 x \log (x)+(-64-18 x) \log ^2(x)+2 x \log ^3(x)-8 \log ^4(x)\right )}{16 x^5+8 x^6+x^7+\left (8 x^5+2 x^6\right ) \log ^2(x)+x^5 \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((20 + 5*x + x*Log[x] + 5*Log[x]^2)/(4 + x + Log[x]^2))*(-128 - 56*x - 6*x^2 + 8*x*Log[x] + (-64 - 18*x
)*Log[x]^2 + 2*x*Log[x]^3 - 8*Log[x]^4))/(16*x^5 + 8*x^6 + x^7 + (8*x^5 + 2*x^6)*Log[x]^2 + x^5*Log[x]^4),x]

[Out]

-8*E^5*Defer[Int][x^(-5 + x/(4 + x + Log[x]^2)), x] + 2*E^5*Defer[Int][x^(-4 + x/(4 + x + Log[x]^2))/(-4 - x -
Log[x]^2), x] + 16*E^5*Defer[Int][x^(-4 + x/(4 + x + Log[x]^2))/(4 + x + Log[x]^2)^2, x] + 4*E^5*Defer[Int][x
^(-3 + x/(4 + x + Log[x]^2))/(4 + x + Log[x]^2)^2, x] - 2*E^5*Defer[Int][(x^(-3 + x/(4 + x + Log[x]^2))*Log[x]
)/(4 + x + Log[x]^2)^2, x] + 2*E^5*Defer[Int][(x^(-4 + x/(4 + x + Log[x]^2))*Log[x])/(4 + x + Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^5 x^{-5+\frac {x}{4+x+\log ^2(x)}} \left (-64-28 x-3 x^2+4 x \log (x)-(32+9 x) \log ^2(x)+x \log ^3(x)-4 \log ^4(x)\right )}{\left (4+x+\log ^2(x)\right )^2} \, dx\\ &=\left (2 e^5\right ) \int \frac {x^{-5+\frac {x}{4+x+\log ^2(x)}} \left (-64-28 x-3 x^2+4 x \log (x)-(32+9 x) \log ^2(x)+x \log ^3(x)-4 \log ^4(x)\right )}{\left (4+x+\log ^2(x)\right )^2} \, dx\\ &=\left (2 e^5\right ) \int \left (-4 x^{-5+\frac {x}{4+x+\log ^2(x)}}-\frac {x^{-4+\frac {x}{4+x+\log ^2(x)}} (-8-2 x+x \log (x))}{\left (4+x+\log ^2(x)\right )^2}+\frac {x^{-4+\frac {x}{4+x+\log ^2(x)}} (-1+\log (x))}{4+x+\log ^2(x)}\right ) \, dx\\ &=-\left (\left (2 e^5\right ) \int \frac {x^{-4+\frac {x}{4+x+\log ^2(x)}} (-8-2 x+x \log (x))}{\left (4+x+\log ^2(x)\right )^2} \, dx\right )+\left (2 e^5\right ) \int \frac {x^{-4+\frac {x}{4+x+\log ^2(x)}} (-1+\log (x))}{4+x+\log ^2(x)} \, dx-\left (8 e^5\right ) \int x^{-5+\frac {x}{4+x+\log ^2(x)}} \, dx\\ &=-\left (\left (2 e^5\right ) \int \left (-\frac {8 x^{-4+\frac {x}{4+x+\log ^2(x)}}}{\left (4+x+\log ^2(x)\right )^2}-\frac {2 x^{-3+\frac {x}{4+x+\log ^2(x)}}}{\left (4+x+\log ^2(x)\right )^2}+\frac {x^{-3+\frac {x}{4+x+\log ^2(x)}} \log (x)}{\left (4+x+\log ^2(x)\right )^2}\right ) \, dx\right )+\left (2 e^5\right ) \int \left (\frac {x^{-4+\frac {x}{4+x+\log ^2(x)}}}{-4-x-\log ^2(x)}+\frac {x^{-4+\frac {x}{4+x+\log ^2(x)}} \log (x)}{4+x+\log ^2(x)}\right ) \, dx-\left (8 e^5\right ) \int x^{-5+\frac {x}{4+x+\log ^2(x)}} \, dx\\ &=\left (2 e^5\right ) \int \frac {x^{-4+\frac {x}{4+x+\log ^2(x)}}}{-4-x-\log ^2(x)} \, dx-\left (2 e^5\right ) \int \frac {x^{-3+\frac {x}{4+x+\log ^2(x)}} \log (x)}{\left (4+x+\log ^2(x)\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x^{-4+\frac {x}{4+x+\log ^2(x)}} \log (x)}{4+x+\log ^2(x)} \, dx+\left (4 e^5\right ) \int \frac {x^{-3+\frac {x}{4+x+\log ^2(x)}}}{\left (4+x+\log ^2(x)\right )^2} \, dx-\left (8 e^5\right ) \int x^{-5+\frac {x}{4+x+\log ^2(x)}} \, dx+\left (16 e^5\right ) \int \frac {x^{-4+\frac {x}{4+x+\log ^2(x)}}}{\left (4+x+\log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.46, size = 20, normalized size = 0.83 \begin {gather*} 2 e^5 x^{-4+\frac {x}{4+x+\log ^2(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((20 + 5*x + x*Log[x] + 5*Log[x]^2)/(4 + x + Log[x]^2))*(-128 - 56*x - 6*x^2 + 8*x*Log[x] + (-64
- 18*x)*Log[x]^2 + 2*x*Log[x]^3 - 8*Log[x]^4))/(16*x^5 + 8*x^6 + x^7 + (8*x^5 + 2*x^6)*Log[x]^2 + x^5*Log[x]^4
),x]

[Out]

2*E^5*x^(-4 + x/(4 + x + Log[x]^2))

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fricas [A]  time = 0.86, size = 31, normalized size = 1.29 \begin {gather*} \frac {2 \, e^{\left (\frac {x \log \relax (x) + 5 \, \log \relax (x)^{2} + 5 \, x + 20}{\log \relax (x)^{2} + x + 4}\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(x)^4+2*x*log(x)^3+(-18*x-64)*log(x)^2+8*x*log(x)-6*x^2-56*x-128)*exp((5*log(x)^2+x*log(x)+20
+5*x)/(log(x)^2+4+x))/(x^5*log(x)^4+(2*x^6+8*x^5)*log(x)^2+x^7+8*x^6+16*x^5),x, algorithm="fricas")

[Out]

2*e^((x*log(x) + 5*log(x)^2 + 5*x + 20)/(log(x)^2 + x + 4))/x^4

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giac [A]  time = 0.61, size = 20, normalized size = 0.83 \begin {gather*} \frac {2 \, x^{\frac {x}{\log \relax (x)^{2} + x + 4}} e^{5}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(x)^4+2*x*log(x)^3+(-18*x-64)*log(x)^2+8*x*log(x)-6*x^2-56*x-128)*exp((5*log(x)^2+x*log(x)+20
+5*x)/(log(x)^2+4+x))/(x^5*log(x)^4+(2*x^6+8*x^5)*log(x)^2+x^7+8*x^6+16*x^5),x, algorithm="giac")

[Out]

2*x^(x/(log(x)^2 + x + 4))*e^5/x^4

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maple [A]  time = 0.03, size = 21, normalized size = 0.88

method	result	size
risch	\(\frac {2 x^{\frac {x}{\ln \relax (x )^{2}+4+x}} {\mathrm e}^{5}}{x^{4}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*ln(x)^4+2*x*ln(x)^3+(-18*x-64)*ln(x)^2+8*x*ln(x)-6*x^2-56*x-128)*exp((5*ln(x)^2+x*ln(x)+20+5*x)/(ln(x)
^2+4+x))/(x^5*ln(x)^4+(2*x^6+8*x^5)*ln(x)^2+x^7+8*x^6+16*x^5),x,method=_RETURNVERBOSE)

[Out]

2/x^4*x^(1/(ln(x)^2+4+x)*x)*exp(5)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {{\left (x \log \relax (x)^{3} - 4 \, \log \relax (x)^{4} - {\left (9 \, x + 32\right )} \log \relax (x)^{2} - 3 \, x^{2} + 4 \, x \log \relax (x) - 28 \, x - 64\right )} e^{\left (\frac {x \log \relax (x) + 5 \, \log \relax (x)^{2} + 5 \, x + 20}{\log \relax (x)^{2} + x + 4}\right )}}{x^{5} \log \relax (x)^{4} + x^{7} + 8 \, x^{6} + 16 \, x^{5} + 2 \, {\left (x^{6} + 4 \, x^{5}\right )} \log \relax (x)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(x)^4+2*x*log(x)^3+(-18*x-64)*log(x)^2+8*x*log(x)-6*x^2-56*x-128)*exp((5*log(x)^2+x*log(x)+20
+5*x)/(log(x)^2+4+x))/(x^5*log(x)^4+(2*x^6+8*x^5)*log(x)^2+x^7+8*x^6+16*x^5),x, algorithm="maxima")

[Out]

2*integrate((x*log(x)^3 - 4*log(x)^4 - (9*x + 32)*log(x)^2 - 3*x^2 + 4*x*log(x) - 28*x - 64)*e^((x*log(x) + 5*
log(x)^2 + 5*x + 20)/(log(x)^2 + x + 4))/(x^5*log(x)^4 + x^7 + 8*x^6 + 16*x^5 + 2*(x^6 + 4*x^5)*log(x)^2), x)

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mupad [B]  time = 4.81, size = 59, normalized size = 2.46 \begin {gather*} \frac {2\,x^{\frac {x}{{\ln \relax (x)}^2+x+4}}\,{\mathrm {e}}^{\frac {20}{{\ln \relax (x)}^2+x+4}}\,{\mathrm {e}}^{\frac {5\,{\ln \relax (x)}^2}{{\ln \relax (x)}^2+x+4}}\,{\mathrm {e}}^{\frac {5\,x}{{\ln \relax (x)}^2+x+4}}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((5*x + 5*log(x)^2 + x*log(x) + 20)/(x + log(x)^2 + 4))*(56*x - 2*x*log(x)^3 + 8*log(x)^4 - 8*x*log(x
) + 6*x^2 + log(x)^2*(18*x + 64) + 128))/(log(x)^2*(8*x^5 + 2*x^6) + x^5*log(x)^4 + 16*x^5 + 8*x^6 + x^7),x)

[Out]

(2*x^(x/(x + log(x)^2 + 4))*exp(20/(x + log(x)^2 + 4))*exp((5*log(x)^2)/(x + log(x)^2 + 4))*exp((5*x)/(x + log
(x)^2 + 4)))/x^4

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sympy [A]  time = 0.51, size = 31, normalized size = 1.29 \begin {gather*} \frac {2 e^{\frac {x \log {\relax (x )} + 5 x + 5 \log {\relax (x )}^{2} + 20}{x + \log {\relax (x )}^{2} + 4}}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*ln(x)**4+2*x*ln(x)**3+(-18*x-64)*ln(x)**2+8*x*ln(x)-6*x**2-56*x-128)*exp((5*ln(x)**2+x*ln(x)+20+
5*x)/(ln(x)**2+4+x))/(x**5*ln(x)**4+(2*x**6+8*x**5)*ln(x)**2+x**7+8*x**6+16*x**5),x)

[Out]

2*exp((x*log(x) + 5*x + 5*log(x)**2 + 20)/(x + log(x)**2 + 4))/x**4

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