3.4.40 \(\int \frac {x+(24 x-5 x^2) \log (x)+6 x \log (x) \log (\frac {5}{\log (x)})}{\sqrt [3]{4-x+\log (\frac {5}{\log (x)})} (e (12-3 x) \log (x)+3 e \log (x) \log (\frac {5}{\log (x)}))} \, dx\) [340]

3.4.40.1 Optimal result

Integrand size = 67, antiderivative size = 23 \begin {dmath*} \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=\frac {x^2}{e \sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )}} \end {dmath*}

output

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

3.4.40.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \begin {dmath*} \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=\frac {x^2}{e \sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )}} \end {dmath*}

input

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]
 

output

x^2/(E*(4 - x + Log[5/Log[x]])^(1/3))
 

3.4.40.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (24 x-5 x^2\right ) \log (x)+x+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{3 e \log (x) \left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {6 \log (x) \log \left (\frac {5}{\log (x)}\right ) x+x+\left (24 x-5 x^2\right ) \log (x)}{\log (x) \left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}dx}{3 e}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\int \left (\frac {6 x \log \left (\frac {5}{\log (x)}\right )}{\left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}-\frac {x (5 x \log (x)-24 \log (x)-1)}{\log (x) \left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}\right )dx}{3 e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-5 \int \frac {x^2}{\left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}dx+24 \int \frac {x}{\left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}dx+\int \frac {x}{\log (x) \left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}dx+6 \int \frac {x \log \left (\frac {5}{\log (x)}\right )}{\left (-x+\log \left (\frac {5}{\log (x)}\right )+4\right )^{4/3}}dx}{3 e}\)

input

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]
 

output

$Aborted
 

3.4.40.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.4.40.4 Maple [F]

\[\int \frac {6 x \ln \left (x \right ) \ln \left (\frac {5}{\ln \left (x \right )}\right )+\left (-5 x^{2}+24 x \right ) \ln \left (x \right )+x}{\left (3 \,{\mathrm e} \ln \left (x \right ) \ln \left (\frac {5}{\ln \left (x \right )}\right )+\left (-3 x +12\right ) {\mathrm e} \ln \left (x \right )\right ) \left (\ln \left (\frac {5}{\ln \left (x \right )}\right )-x +4\right )^{\frac {1}{3}}}d x\]

input

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)
 

output

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)
 

3.4.40.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \begin {dmath*} \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=-\frac {x^{2} {\left (-x + \log \left (\frac {5}{\log \left (x\right )}\right ) + 4\right )}^{\frac {2}{3}}}{{\left (x - 4\right )} e - e \log \left (\frac {5}{\log \left (x\right )}\right )} \end {dmath*}

input

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)*log(x))/(log(5/log(x))-x+4)^(1/3),x, alg 
orithm=\
 

output

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

3.4.40.6 Sympy [F(-1)]

Timed out. \begin {dmath*} \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=\text {Timed out} \end {dmath*}

input

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

output

Timed out
 

3.4.40.7 Maxima [F]

\begin {dmath*} \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=\int { -\frac {6 \, x \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right ) - {\left (5 \, x^{2} - 24 \, x\right )} \log \left (x\right ) + x}{3 \, {\left ({\left (x - 4\right )} e \log \left (x\right ) - e \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right )\right )} {\left (-x + \log \left (\frac {5}{\log \left (x\right )}\right ) + 4\right )}^{\frac {1}{3}}} \,d x } \end {dmath*}

input

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)*log(x))/(log(5/log(x))-x+4)^(1/3),x, alg 
orithm=\
 

output

-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)
 

3.4.40.8 Giac [F]

\begin {dmath*} \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=\int { -\frac {6 \, x \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right ) - {\left (5 \, x^{2} - 24 \, x\right )} \log \left (x\right ) + x}{3 \, {\left ({\left (x - 4\right )} e \log \left (x\right ) - e \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right )\right )} {\left (-x + \log \left (\frac {5}{\log \left (x\right )}\right ) + 4\right )}^{\frac {1}{3}}} \,d x } \end {dmath*}

input

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)*log(x))/(log(5/log(x))-x+4)^(1/3),x, alg 
orithm=\
 

output

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)
 

3.4.40.9 Mupad [B] (verification not implemented)

Time = 13.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \begin {dmath*} \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=\frac {x^2\,{\mathrm {e}}^{-1}}{{\left (\ln \left (\frac {5}{\ln \left (x\right )}\right )-x+4\right )}^{1/3}} \end {dmath*}

input

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)
 

output

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