3.27.72 \(\int \frac {1}{2} e^{-3+(-5 e^{x/2} x+20 x^5) \log (x)} (-10 e^{x/2}+40 x^4+(e^{x/2} (-10-5 x)+200 x^4) \log (x)) \, dx\)

Optimal. Leaf size=24 \[ e^{-3+5 x \left (-e^{x/2}+4 x^4\right ) \log (x)} \]

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Rubi [F]  time = 3.35, 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 {1}{2} e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} \left (-10 e^{x/2}+40 x^4+\left (e^{x/2} (-10-5 x)+200 x^4\right ) \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-3 + (-5*E^(x/2)*x + 20*x^5)*Log[x])*(-10*E^(x/2) + 40*x^4 + (E^(x/2)*(-10 - 5*x) + 200*x^4)*Log[x]))/
2,x]

[Out]

-5*Defer[Int][E^((-6 + x - 10*E^(x/2)*x*Log[x] + 40*x^5*Log[x])/2), x] + 20*Defer[Int][E^(-3 + (-5*E^(x/2)*x +
20*x^5)*Log[x])*x^4, x] - 5*Defer[Int][E^((-6 + x - 10*E^(x/2)*x*Log[x] + 40*x^5*Log[x])/2)*Log[x], x] - (5*D
efer[Int][E^((-6 + x - 10*E^(x/2)*x*Log[x] + 40*x^5*Log[x])/2)*x*Log[x], x])/2 + 100*Defer[Int][E^(-3 + (-5*E^
(x/2)*x + 20*x^5)*Log[x])*x^4*Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} \left (-10 e^{x/2}+40 x^4+\left (e^{x/2} (-10-5 x)+200 x^4\right ) \log (x)\right ) \, dx\\ &=\frac {1}{2} \int \left (-10 e^{-3+\frac {x}{2}+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)}+40 e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4-5 e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} \left (2 e^{x/2}+e^{x/2} x-40 x^4\right ) \log (x)\right ) \, dx\\ &=-\left (\frac {5}{2} \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} \left (2 e^{x/2}+e^{x/2} x-40 x^4\right ) \log (x) \, dx\right )-5 \int e^{-3+\frac {x}{2}+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} \, dx+20 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \, dx\\ &=-\left (\frac {5}{2} \int \left (-40 e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \log (x)+e^{-3+\frac {x}{2}+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} (2+x) \log (x)\right ) \, dx\right )-5 \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) \, dx+20 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \, dx\\ &=-\left (\frac {5}{2} \int e^{-3+\frac {x}{2}+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} (2+x) \log (x) \, dx\right )-5 \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) \, dx+20 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \, dx+100 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \log (x) \, dx\\ &=-\left (\frac {5}{2} \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) (2+x) \log (x) \, dx\right )-5 \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) \, dx+20 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \, dx+100 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \log (x) \, dx\\ &=-\left (\frac {5}{2} \int \left (2 \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) \log (x)+\exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) x \log (x)\right ) \, dx\right )-5 \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) \, dx+20 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \, dx+100 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \log (x) \, dx\\ &=-\left (\frac {5}{2} \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) x \log (x) \, dx\right )-5 \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) \, dx-5 \int \exp \left (\frac {1}{2} \left (-6+x-10 e^{x/2} x \log (x)+40 x^5 \log (x)\right )\right ) \log (x) \, dx+20 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \, dx+100 \int e^{-3+\left (-5 e^{x/2} x+20 x^5\right ) \log (x)} x^4 \log (x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 24, normalized size = 1.00 \begin {gather*} \frac {x^{5 x \left (-e^{x/2}+4 x^4\right )}}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-3 + (-5*E^(x/2)*x + 20*x^5)*Log[x])*(-10*E^(x/2) + 40*x^4 + (E^(x/2)*(-10 - 5*x) + 200*x^4)*Log
[x]))/2,x]

[Out]

x^(5*x*(-E^(x/2) + 4*x^4))/E^3

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fricas [A]  time = 0.71, size = 20, normalized size = 0.83 \begin {gather*} e^{\left (5 \, {\left (4 \, x^{5} - x e^{\left (\frac {1}{2} \, x\right )}\right )} \log \relax (x) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-5*x-10)*exp(1/2*x)+200*x^4)*log(x)-10*exp(1/2*x)+40*x^4)*exp((-5*x*exp(1/2*x)+20*x^5)*log(x)
-3),x, algorithm="fricas")

[Out]

e^(5*(4*x^5 - x*e^(1/2*x))*log(x) - 3)

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giac [A]  time = 0.32, size = 19, normalized size = 0.79 \begin {gather*} e^{\left (20 \, x^{5} \log \relax (x) - 5 \, x e^{\left (\frac {1}{2} \, x\right )} \log \relax (x) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-5*x-10)*exp(1/2*x)+200*x^4)*log(x)-10*exp(1/2*x)+40*x^4)*exp((-5*x*exp(1/2*x)+20*x^5)*log(x)
-3),x, algorithm="giac")

[Out]

e^(20*x^5*log(x) - 5*x*e^(1/2*x)*log(x) - 3)

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maple [A]  time = 0.07, size = 19, normalized size = 0.79

method	result	size
risch	\(x^{-5 x \left (-4 x^{4}+{\mathrm e}^{\frac {x}{2}}\right )} {\mathrm e}^{-3}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((-5*x-10)*exp(1/2*x)+200*x^4)*ln(x)-10*exp(1/2*x)+40*x^4)*exp((-5*x*exp(1/2*x)+20*x^5)*ln(x)-3),x,me
thod=_RETURNVERBOSE)

[Out]

x^(-5*x*(-4*x^4+exp(1/2*x)))*exp(-3)

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maxima [A]  time = 0.82, size = 19, normalized size = 0.79 \begin {gather*} e^{\left (20 \, x^{5} \log \relax (x) - 5 \, x e^{\left (\frac {1}{2} \, x\right )} \log \relax (x) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-5*x-10)*exp(1/2*x)+200*x^4)*log(x)-10*exp(1/2*x)+40*x^4)*exp((-5*x*exp(1/2*x)+20*x^5)*log(x)
-3),x, algorithm="maxima")

[Out]

e^(20*x^5*log(x) - 5*x*e^(1/2*x)*log(x) - 3)

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mupad [B]  time = 1.72, size = 18, normalized size = 0.75 \begin {gather*} x^{20\,x^5-5\,x\,{\mathrm {e}}^{x/2}}\,{\mathrm {e}}^{-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- log(x)*(5*x*exp(x/2) - 20*x^5) - 3)*(10*exp(x/2) + log(x)*(exp(x/2)*(5*x + 10) - 200*x^4) - 40*x^4
))/2,x)

[Out]

x^(20*x^5 - 5*x*exp(x/2))*exp(-3)

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sympy [A]  time = 0.53, size = 19, normalized size = 0.79 \begin {gather*} e^{\left (20 x^{5} - 5 x e^{\frac {x}{2}}\right ) \log {\relax (x )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-5*x-10)*exp(1/2*x)+200*x**4)*ln(x)-10*exp(1/2*x)+40*x**4)*exp((-5*x*exp(1/2*x)+20*x**5)*ln(x
)-3),x)

[Out]

exp((20*x**5 - 5*x*exp(x/2))*log(x) - 3)

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