3.94.40 \(\int \frac {x+(24 x-5 x^2) \log (x)+6 x \log (x) \log (\frac {5}{\log (x)})}{\sqrt [3]{4-x+\log (\frac {5}{\log (x)})} (e (12-3 x) \log (x)+3 e \log (x) \log (\frac {5}{\log (x)}))} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^2}{e \sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )}} \]

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Rubi [F]  time = 1.99, 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 {x+\left (24 x-5 x^2\right ) \log (x)+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{\sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )} \left (e (12-3 x) \log (x)+3 e \log (x) \log \left (\frac {5}{\log (x)}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x + (24*x - 5*x^2)*Log[x] + 6*x*Log[x]*Log[5/Log[x]])/((4 - x + Log[5/Log[x]])^(1/3)*(E*(12 - 3*x)*Log[x]
+ 3*E*Log[x]*Log[5/Log[x]])),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (1-\log (x) \left (-24+5 x-6 \log \left (\frac {5}{\log (x)}\right )\right )\right )}{3 e \log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx\\ &=\frac {\int \frac {x \left (1-\log (x) \left (-24+5 x-6 \log \left (\frac {5}{\log (x)}\right )\right )\right )}{\log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{3 e}\\ &=\frac {\int \left (-\frac {x (-1-24 \log (x)+5 x \log (x))}{\log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}}+\frac {6 x \log \left (\frac {5}{\log (x)}\right )}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}}\right ) \, dx}{3 e}\\ &=-\frac {\int \frac {x (-1-24 \log (x)+5 x \log (x))}{\log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{3 e}+\frac {2 \int \frac {x \log \left (\frac {5}{\log (x)}\right )}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{e}\\ &=-\frac {\int \left (\frac {x (-24+5 x)}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}}-\frac {x}{\log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}}\right ) \, dx}{3 e}+\frac {2 \int \frac {x \log \left (\frac {5}{\log (x)}\right )}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{e}\\ &=-\frac {\int \frac {x (-24+5 x)}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{3 e}+\frac {\int \frac {x}{\log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{3 e}+\frac {2 \int \frac {x \log \left (\frac {5}{\log (x)}\right )}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{e}\\ &=\frac {\int \frac {x}{\log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{3 e}-\frac {\int \left (-\frac {24 x}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}}+\frac {5 x^2}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}}\right ) \, dx}{3 e}+\frac {2 \int \frac {x \log \left (\frac {5}{\log (x)}\right )}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{e}\\ &=\frac {\int \frac {x}{\log (x) \left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{3 e}-\frac {5 \int \frac {x^2}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{3 e}+\frac {2 \int \frac {x \log \left (\frac {5}{\log (x)}\right )}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{e}+\frac {8 \int \frac {x}{\left (4-x+\log \left (\frac {5}{\log (x)}\right )\right )^{4/3}} \, dx}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^2}{e \sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + (24*x - 5*x^2)*Log[x] + 6*x*Log[x]*Log[5/Log[x]])/((4 - x + Log[5/Log[x]])^(1/3)*(E*(12 - 3*x)*
Log[x] + 3*E*Log[x]*Log[5/Log[x]])),x]

[Out]

x^2/(E*(4 - x + Log[5/Log[x]])^(1/3))

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fricas [A]  time = 0.59, size = 39, normalized size = 1.70 \begin {gather*} -\frac {x^{2} {\left (-x + \log \left (\frac {5}{\log \relax (x)}\right ) + 4\right )}^{\frac {2}{3}}}{{\left (x - 4\right )} e - e \log \left (\frac {5}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(x)*log(5/log(x))+(-5*x^2+24*x)*log(x)+x)/(3*exp(1)*log(x)*log(5/log(x))+(-3*x+12)*exp(1)*lo
g(x))/(log(5/log(x))-x+4)^(1/3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {6 \, x \log \relax (x) \log \left (\frac {5}{\log \relax (x)}\right ) - {\left (5 \, x^{2} - 24 \, x\right )} \log \relax (x) + x}{3 \, {\left ({\left (x - 4\right )} e \log \relax (x) - e \log \relax (x) \log \left (\frac {5}{\log \relax (x)}\right )\right )} {\left (-x + \log \left (\frac {5}{\log \relax (x)}\right ) + 4\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(x)*log(5/log(x))+(-5*x^2+24*x)*log(x)+x)/(3*exp(1)*log(x)*log(5/log(x))+(-3*x+12)*exp(1)*lo
g(x))/(log(5/log(x))-x+4)^(1/3),x, algorithm="giac")

[Out]

integrate(-1/3*(6*x*log(x)*log(5/log(x)) - (5*x^2 - 24*x)*log(x) + x)/(((x - 4)*e*log(x) - e*log(x)*log(5/log(
x)))*(-x + log(5/log(x)) + 4)^(1/3)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {6 x \ln \relax (x ) \ln \left (\frac {5}{\ln \relax (x )}\right )+\left (-5 x^{2}+24 x \right ) \ln \relax (x )+x}{\left (3 \,{\mathrm e} \ln \relax (x ) \ln \left (\frac {5}{\ln \relax (x )}\right )+\left (-3 x +12\right ) {\mathrm e} \ln \relax (x )\right ) \left (\ln \left (\frac {5}{\ln \relax (x )}\right )-x +4\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*ln(x)*ln(5/ln(x))+(-5*x^2+24*x)*ln(x)+x)/(3*exp(1)*ln(x)*ln(5/ln(x))+(-3*x+12)*exp(1)*ln(x))/(ln(5/ln
(x))-x+4)^(1/3),x)

[Out]

int((6*x*ln(x)*ln(5/ln(x))+(-5*x^2+24*x)*ln(x)+x)/(3*exp(1)*ln(x)*ln(5/ln(x))+(-3*x+12)*exp(1)*ln(x))/(ln(5/ln
(x))-x+4)^(1/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{3} \, \int \frac {6 \, x \log \relax (x) \log \left (\frac {5}{\log \relax (x)}\right ) - {\left (5 \, x^{2} - 24 \, x\right )} \log \relax (x) + x}{{\left ({\left (x - 4\right )} e \log \relax (x) - e \log \relax (x) \log \left (\frac {5}{\log \relax (x)}\right )\right )} {\left (-x + \log \left (\frac {5}{\log \relax (x)}\right ) + 4\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(x)*log(5/log(x))+(-5*x^2+24*x)*log(x)+x)/(3*exp(1)*log(x)*log(5/log(x))+(-3*x+12)*exp(1)*lo
g(x))/(log(5/log(x))-x+4)^(1/3),x, algorithm="maxima")

[Out]

-1/3*integrate((6*x*log(x)*log(5/log(x)) - (5*x^2 - 24*x)*log(x) + x)/(((x - 4)*e*log(x) - e*log(x)*log(5/log(
x)))*(-x + log(5/log(x)) + 4)^(1/3)), x)

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mupad [B]  time = 7.78, size = 20, normalized size = 0.87 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{-1}}{{\left (\ln \left (\frac {5}{\ln \relax (x)}\right )-x+4\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + log(x)*(24*x - 5*x^2) + 6*x*log(5/log(x))*log(x))/((3*log(5/log(x))*exp(1)*log(x) - exp(1)*log(x)*(3*
x - 12))*(log(5/log(x)) - x + 4)^(1/3)),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*ln(x)*ln(5/ln(x))+(-5*x**2+24*x)*ln(x)+x)/(3*exp(1)*ln(x)*ln(5/ln(x))+(-3*x+12)*exp(1)*ln(x))/(
ln(5/ln(x))-x+4)**(1/3),x)

[Out]

Timed out

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