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

Optimal result

Integrand size = 67, antiderivative size = 23 \[ \int \frac {x+\left (24 x-5 x^2\right ) \log (x)+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{\sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )} \left (e (12-3 x) \log (x)+3 e \log (x) \log \left (\frac {5}{\log (x)}\right )\right )} \, dx=\frac {x^2}{e \sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )}} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x+\left (24 x-5 x^2\right ) \log (x)+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{\sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )} \left (e (12-3 x) \log (x)+3 e \log (x) \log \left (\frac {5}{\log (x)}\right )\right )} \, dx=\frac {x^2}{e \sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )}} \] 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))
 

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
 

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)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {x+\left (24 x-5 x^2\right ) \log (x)+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{\sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )} \left (e (12-3 x) \log (x)+3 e \log (x) \log \left (\frac {5}{\log (x)}\right )\right )} \, dx=-\frac {x^{2} {\left (-x + \log \left (\frac {5}{\log \left (x\right )}\right ) + 4\right )}^{\frac {2}{3}}}{{\left (x - 4\right )} e - e \log \left (\frac {5}{\log \left (x\right )}\right )} \] 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="fricas")
 

Output:

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

Sympy [F(-1)]

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

Maxima [F]

\[ \int \frac {x+\left (24 x-5 x^2\right ) \log (x)+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{\sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )} \left (e (12-3 x) \log (x)+3 e \log (x) \log \left (\frac {5}{\log (x)}\right )\right )} \, dx=\int { -\frac {6 \, x \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right ) - {\left (5 \, x^{2} - 24 \, x\right )} \log \left (x\right ) + x}{3 \, {\left ({\left (x - 4\right )} e \log \left (x\right ) - e \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right )\right )} {\left (-x + \log \left (\frac {5}{\log \left (x\right )}\right ) + 4\right )}^{\frac {1}{3}}} \,d x } \] 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="maxima")
 

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)
 

Giac [F]

\[ \int \frac {x+\left (24 x-5 x^2\right ) \log (x)+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{\sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )} \left (e (12-3 x) \log (x)+3 e \log (x) \log \left (\frac {5}{\log (x)}\right )\right )} \, dx=\int { -\frac {6 \, x \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right ) - {\left (5 \, x^{2} - 24 \, x\right )} \log \left (x\right ) + x}{3 \, {\left ({\left (x - 4\right )} e \log \left (x\right ) - e \log \left (x\right ) \log \left (\frac {5}{\log \left (x\right )}\right )\right )} {\left (-x + \log \left (\frac {5}{\log \left (x\right )}\right ) + 4\right )}^{\frac {1}{3}}} \,d x } \] 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="giac")
 

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)
 

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {x+\left (24 x-5 x^2\right ) \log (x)+6 x \log (x) \log \left (\frac {5}{\log (x)}\right )}{\sqrt [3]{4-x+\log \left (\frac {5}{\log (x)}\right )} \left (e (12-3 x) \log (x)+3 e \log (x) \log \left (\frac {5}{\log (x)}\right )\right )} \, dx=\frac {x^2\,{\mathrm {e}}^{-1}}{{\left (\ln \left (\frac {5}{\ln \left (x\right )}\right )-x+4\right )}^{1/3}} \] 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)
 

Reduce [F]

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

int((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)
 

Output:

( - 5*int(x**2/((log(5/log(x)) - x + 4)**(1/3)*log(5/log(x)) - (log(5/log( 
x)) - x + 4)**(1/3)*x + 4*(log(5/log(x)) - x + 4)**(1/3)),x) + 6*int((log( 
5/log(x))*x)/((log(5/log(x)) - x + 4)**(1/3)*log(5/log(x)) - (log(5/log(x) 
) - x + 4)**(1/3)*x + 4*(log(5/log(x)) - x + 4)**(1/3)),x) + int(x/((log(5 
/log(x)) - x + 4)**(1/3)*log(5/log(x))*log(x) - (log(5/log(x)) - x + 4)**( 
1/3)*log(x)*x + 4*(log(5/log(x)) - x + 4)**(1/3)*log(x)),x) + 24*int(x/((l 
og(5/log(x)) - x + 4)**(1/3)*log(5/log(x)) - (log(5/log(x)) - x + 4)**(1/3 
)*x + 4*(log(5/log(x)) - x + 4)**(1/3)),x))/(3*e)