3.89.69 \(\int \frac {e^{\frac {1}{-400 x^5+(2000 x^4+400 x^5) \log (x)}} (-5+4 x+(-20-5 x) \log (x))}{200 x^7+(-2000 x^6-400 x^7) \log (x)+(5000 x^5+2000 x^6+200 x^7) \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ 2 e^{-\frac {1}{400 x^4 (x-(5+x) \log (x))}} \]

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Rubi [F]  time = 5.22, 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 {1}{-400 x^5+\left (2000 x^4+400 x^5\right ) \log (x)}} (-5+4 x+(-20-5 x) \log (x))}{200 x^7+\left (-2000 x^6-400 x^7\right ) \log (x)+\left (5000 x^5+2000 x^6+200 x^7\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-400*x^5 + (2000*x^4 + 400*x^5)*Log[x])^(-1)*(-5 + 4*x + (-20 - 5*x)*Log[x]))/(200*x^7 + (-2000*x^6 -
400*x^7)*Log[x] + (5000*x^5 + 2000*x^6 + 200*x^7)*Log[x]^2),x]

[Out]

-1/40*Defer[Int][E^(1/(400*x^4*(-x + (5 + x)*Log[x])))/(x^5*(-x + 5*Log[x] + x*Log[x])^2), x] - Defer[Int][E^(
1/(400*x^4*(-x + (5 + x)*Log[x])))/(x^3*(-x + 5*Log[x] + x*Log[x])^2), x]/1000 + Defer[Int][E^(1/(400*x^4*(-x
+ (5 + x)*Log[x])))/(x^2*(-x + 5*Log[x] + x*Log[x])^2), x]/5000 - Defer[Int][E^(1/(400*x^4*(-x + (5 + x)*Log[x
])))/(x*(-x + 5*Log[x] + x*Log[x])^2), x]/25000 + Defer[Int][E^(1/(400*x^4*(-x + (5 + x)*Log[x])))/((5 + x)*(-
x + 5*Log[x] + x*Log[x])^2), x]/25000 - Defer[Int][E^(1/(400*x^4*(-x + (5 + x)*Log[x])))/(x^5*(-x + 5*Log[x] +
x*Log[x])), x]/50 - Defer[Int][E^(1/(400*x^4*(-x + (5 + x)*Log[x])))/(x^4*(-x + 5*Log[x] + x*Log[x])), x]/100
0 + Defer[Int][E^(1/(400*x^4*(-x + (5 + x)*Log[x])))/(x^3*(-x + 5*Log[x] + x*Log[x])), x]/5000 - Defer[Int][E^
(1/(400*x^4*(-x + (5 + x)*Log[x])))/(x^2*(-x + 5*Log[x] + x*Log[x])), x]/25000 + Defer[Int][E^(1/(400*x^4*(-x
+ (5 + x)*Log[x])))/(x*(-x + 5*Log[x] + x*Log[x])), x]/125000 - Defer[Int][E^(1/(400*x^4*(-x + (5 + x)*Log[x])
))/((5 + x)*(-x + 5*Log[x] + x*Log[x])), x]/125000

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}} (-5+4 x-5 (4+x) \log (x))}{200 x^5 (x-(5+x) \log (x))^2} \, dx\\ &=\frac {1}{200} \int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}} (-5+4 x-5 (4+x) \log (x))}{x^5 (x-(5+x) \log (x))^2} \, dx\\ &=\frac {1}{200} \int \left (\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}} \left (-25-5 x-x^2\right )}{x^5 (5+x) (-x+5 \log (x)+x \log (x))^2}-\frac {5 e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}} (4+x)}{x^5 (5+x) (-x+5 \log (x)+x \log (x))}\right ) \, dx\\ &=\frac {1}{200} \int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}} \left (-25-5 x-x^2\right )}{x^5 (5+x) (-x+5 \log (x)+x \log (x))^2} \, dx-\frac {1}{40} \int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}} (4+x)}{x^5 (5+x) (-x+5 \log (x)+x \log (x))} \, dx\\ &=\frac {1}{200} \int \left (-\frac {5 e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^5 (-x+5 \log (x)+x \log (x))^2}-\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{5 x^3 (-x+5 \log (x)+x \log (x))^2}+\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{25 x^2 (-x+5 \log (x)+x \log (x))^2}-\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{125 x (-x+5 \log (x)+x \log (x))^2}+\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{125 (5+x) (-x+5 \log (x)+x \log (x))^2}\right ) \, dx-\frac {1}{40} \int \left (\frac {4 e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{5 x^5 (-x+5 \log (x)+x \log (x))}+\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{25 x^4 (-x+5 \log (x)+x \log (x))}-\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{125 x^3 (-x+5 \log (x)+x \log (x))}+\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{625 x^2 (-x+5 \log (x)+x \log (x))}-\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{3125 x (-x+5 \log (x)+x \log (x))}+\frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{3125 (5+x) (-x+5 \log (x)+x \log (x))}\right ) \, dx\\ &=\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x (-x+5 \log (x)+x \log (x))} \, dx}{125000}-\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{(5+x) (-x+5 \log (x)+x \log (x))} \, dx}{125000}-\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x (-x+5 \log (x)+x \log (x))^2} \, dx}{25000}+\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{(5+x) (-x+5 \log (x)+x \log (x))^2} \, dx}{25000}-\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^2 (-x+5 \log (x)+x \log (x))} \, dx}{25000}+\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^2 (-x+5 \log (x)+x \log (x))^2} \, dx}{5000}+\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^3 (-x+5 \log (x)+x \log (x))} \, dx}{5000}-\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^3 (-x+5 \log (x)+x \log (x))^2} \, dx}{1000}-\frac {\int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^4 (-x+5 \log (x)+x \log (x))} \, dx}{1000}-\frac {1}{50} \int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^5 (-x+5 \log (x)+x \log (x))} \, dx-\frac {1}{40} \int \frac {e^{\frac {1}{400 x^4 (-x+(5+x) \log (x))}}}{x^5 (-x+5 \log (x)+x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.34, size = 25, normalized size = 1.14 \begin {gather*} 2 e^{\frac {1}{400 x^4 (-x+5 \log (x)+x \log (x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-400*x^5 + (2000*x^4 + 400*x^5)*Log[x])^(-1)*(-5 + 4*x + (-20 - 5*x)*Log[x]))/(200*x^7 + (-2000*
x^6 - 400*x^7)*Log[x] + (5000*x^5 + 2000*x^6 + 200*x^7)*Log[x]^2),x]

[Out]

2*E^(1/(400*x^4*(-x + 5*Log[x] + x*Log[x])))

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fricas [A]  time = 0.64, size = 24, normalized size = 1.09 \begin {gather*} 2 \, e^{\left (-\frac {1}{400 \, {\left (x^{5} - {\left (x^{5} + 5 \, x^{4}\right )} \log \relax (x)\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-20)*log(x)+4*x-5)/((200*x^7+2000*x^6+5000*x^5)*log(x)^2+(-400*x^7-2000*x^6)*log(x)+200*x^7)/e
xp(-1/((400*x^5+2000*x^4)*log(x)-400*x^5)),x, algorithm="fricas")

[Out]

2*e^(-1/400/(x^5 - (x^5 + 5*x^4)*log(x)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (5 \, {\left (x + 4\right )} \log \relax (x) - 4 \, x + 5\right )} e^{\left (-\frac {1}{400 \, {\left (x^{5} - {\left (x^{5} + 5 \, x^{4}\right )} \log \relax (x)\right )}}\right )}}{200 \, {\left (x^{7} + {\left (x^{7} + 10 \, x^{6} + 25 \, x^{5}\right )} \log \relax (x)^{2} - 2 \, {\left (x^{7} + 5 \, x^{6}\right )} \log \relax (x)\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-20)*log(x)+4*x-5)/((200*x^7+2000*x^6+5000*x^5)*log(x)^2+(-400*x^7-2000*x^6)*log(x)+200*x^7)/e
xp(-1/((400*x^5+2000*x^4)*log(x)-400*x^5)),x, algorithm="giac")

[Out]

integrate(-1/200*(5*(x + 4)*log(x) - 4*x + 5)*e^(-1/400/(x^5 - (x^5 + 5*x^4)*log(x)))/(x^7 + (x^7 + 10*x^6 + 2
5*x^5)*log(x)^2 - 2*(x^7 + 5*x^6)*log(x)), x)

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maple [A]  time = 0.03, size = 23, normalized size = 1.05

method	result	size
risch	\(2 \,{\mathrm e}^{\frac {1}{400 x^{4} \left (x \ln \relax (x )+5 \ln \relax (x )-x \right )}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x-20)*ln(x)+4*x-5)/((200*x^7+2000*x^6+5000*x^5)*ln(x)^2+(-400*x^7-2000*x^6)*ln(x)+200*x^7)/exp(-1/((4
00*x^5+2000*x^4)*ln(x)-400*x^5)),x,method=_RETURNVERBOSE)

[Out]

2*exp(1/400/x^4/(x*ln(x)+5*ln(x)-x))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-20)*log(x)+4*x-5)/((200*x^7+2000*x^6+5000*x^5)*log(x)^2+(-400*x^7-2000*x^6)*log(x)+200*x^7)/e
xp(-1/((400*x^5+2000*x^4)*log(x)-400*x^5)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [B]  time = 5.72, size = 25, normalized size = 1.14 \begin {gather*} 2\,{\mathrm {e}}^{\frac {1}{2000\,x^4\,\ln \relax (x)+400\,x^5\,\ln \relax (x)-400\,x^5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1/(log(x)*(2000*x^4 + 400*x^5) - 400*x^5))*(log(x)*(5*x + 20) - 4*x + 5))/(log(x)^2*(5000*x^5 + 2000
*x^6 + 200*x^7) - log(x)*(2000*x^6 + 400*x^7) + 200*x^7),x)

[Out]

2*exp(1/(2000*x^4*log(x) + 400*x^5*log(x) - 400*x^5))

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sympy [A]  time = 0.63, size = 22, normalized size = 1.00 \begin {gather*} 2 e^{\frac {1}{- 400 x^{5} + \left (400 x^{5} + 2000 x^{4}\right ) \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x-20)*ln(x)+4*x-5)/((200*x**7+2000*x**6+5000*x**5)*ln(x)**2+(-400*x**7-2000*x**6)*ln(x)+200*x**
7)/exp(-1/((400*x**5+2000*x**4)*ln(x)-400*x**5)),x)

[Out]

2*exp(1/(-400*x**5 + (400*x**5 + 2000*x**4)*log(x)))

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