\(\int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+(4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+(-2187+162 x^2-24 x^3+x^4) \log (\frac {5}{3+x})) \log (\frac {162-118 x+20 x^2-x^3+(-81+18 x-x^2) \log (\frac {5}{3+x})}{81-18 x+x^2})}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+(-2187+162 x^2-24 x^3+x^4) \log (\frac {5}{3+x})} \, dx\) [1758]

Optimal result

Integrand size = 166, antiderivative size = 28 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=x \log \left (2-x-\frac {x}{(9-x)^2}-\log \left (\frac {5}{3+x}\right )\right ) \] Output:

x*ln(-x-x/(9-x)^2-ln(5/(3+x))+2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=x \log \left (-\frac {-162+118 x-20 x^2+x^3+(-9+x)^2 \log \left (\frac {5}{3+x}\right )}{(-9+x)^2}\right ) \] Input:

Integrate[(-1485*x - 255*x^2 + 188*x^3 - 25*x^4 + x^5 + (4374 - 2214*x - 3 
30*x^2 + 211*x^3 - 26*x^4 + x^5 + (-2187 + 162*x^2 - 24*x^3 + x^4)*Log[5/( 
3 + x)])*Log[(162 - 118*x + 20*x^2 - x^3 + (-81 + 18*x - x^2)*Log[5/(3 + x 
)])/(81 - 18*x + x^2)])/(4374 - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 
+ (-2187 + 162*x^2 - 24*x^3 + x^4)*Log[5/(3 + x)]),x]
 

Output:

x*Log[-((-162 + 118*x - 20*x^2 + x^3 + (-9 + x)^2*Log[5/(3 + x)])/(-9 + x) 
^2)]
 

Rubi [A] (verified)

Time = 9.90 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {7293, 2009}

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.

\(\displaystyle \int \frac {x^5-25 x^4+188 x^3-255 x^2+\left (x^5-26 x^4+211 x^3-330 x^2+\left (x^4-24 x^3+162 x^2-2187\right ) \log \left (\frac {5}{x+3}\right )-2214 x+4374\right ) \log \left (\frac {-x^3+20 x^2+\left (-x^2+18 x-81\right ) \log \left (\frac {5}{x+3}\right )-118 x+162}{x^2-18 x+81}\right )-1485 x}{x^5-26 x^4+211 x^3-330 x^2+\left (x^4-24 x^3+162 x^2-2187\right ) \log \left (\frac {5}{x+3}\right )-2214 x+4374} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {188 x^3}{(x-9) (x+3) \left (x^3-20 x^2+x^2 \log \left (\frac {5}{x+3}\right )+118 x-18 x \log \left (\frac {5}{x+3}\right )+81 \log \left (\frac {5}{x+3}\right )-162\right )}-\frac {255 x^2}{(x-9) (x+3) \left (x^3-20 x^2+x^2 \log \left (\frac {5}{x+3}\right )+118 x-18 x \log \left (\frac {5}{x+3}\right )+81 \log \left (\frac {5}{x+3}\right )-162\right )}-\frac {1485 x}{(x-9) (x+3) \left (x^3-20 x^2+x^2 \log \left (\frac {5}{x+3}\right )+118 x-18 x \log \left (\frac {5}{x+3}\right )+81 \log \left (\frac {5}{x+3}\right )-162\right )}+\log \left (-\frac {x^3-20 x^2+118 x+(x-9)^2 \log \left (\frac {5}{x+3}\right )-162}{(x-9)^2}\right )+\frac {x^5}{(x-9) (x+3) \left (x^3-20 x^2+x^2 \log \left (\frac {5}{x+3}\right )+118 x-18 x \log \left (\frac {5}{x+3}\right )+81 \log \left (\frac {5}{x+3}\right )-162\right )}-\frac {25 x^4}{(x-9) (x+3) \left (x^3-20 x^2+x^2 \log \left (\frac {5}{x+3}\right )+118 x-18 x \log \left (\frac {5}{x+3}\right )+81 \log \left (\frac {5}{x+3}\right )-162\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x \log \left (\frac {-x^3+20 x^2-118 x-(9-x)^2 \log \left (\frac {5}{x+3}\right )+162}{(9-x)^2}\right )\)

Input:

Int[(-1485*x - 255*x^2 + 188*x^3 - 25*x^4 + x^5 + (4374 - 2214*x - 330*x^2 
 + 211*x^3 - 26*x^4 + x^5 + (-2187 + 162*x^2 - 24*x^3 + x^4)*Log[5/(3 + x) 
])*Log[(162 - 118*x + 20*x^2 - x^3 + (-81 + 18*x - x^2)*Log[5/(3 + x)])/(8 
1 - 18*x + x^2)])/(4374 - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 + (-21 
87 + 162*x^2 - 24*x^3 + x^4)*Log[5/(3 + x)]),x]
 

Output:

x*Log[(162 - 118*x + 20*x^2 - x^3 - (9 - x)^2*Log[5/(3 + x)])/(9 - x)^2]
 

Defintions of rubi rules used

rule 2009

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

rule 7293

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

Maple [A] (verified)

Time = 9.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75

method	result	size
parallelrisch	\(\ln \left (\frac {\left (-x^{2}+18 x -81\right ) \ln \left (\frac {5}{3+x}\right )-x^{3}+20 x^{2}-118 x +162}{x^{2}-18 x +81}\right ) x\)	\(49\)
default	\(\ln \left (\frac {-x^{2} \ln \left (5\right )-x^{2} \ln \left (\frac {1}{3+x}\right )-x^{3}+18 x \ln \left (5\right )+18 x \ln \left (\frac {1}{3+x}\right )+20 x^{2}-81 \ln \left (5\right )-81 \ln \left (\frac {1}{3+x}\right )-118 x +162}{x^{2}-18 x +81}\right ) x\)	\(74\)
parts	\(\ln \left (\frac {-x^{2} \ln \left (5\right )-x^{2} \ln \left (\frac {1}{3+x}\right )-x^{3}+18 x \ln \left (5\right )+18 x \ln \left (\frac {1}{3+x}\right )+20 x^{2}-81 \ln \left (5\right )-81 \ln \left (\frac {1}{3+x}\right )-118 x +162}{x^{2}-18 x +81}\right ) x\)	\(74\)

Input:

int((((x^4-24*x^3+162*x^2-2187)*ln(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-221 
4*x+4374)*ln(((-x^2+18*x-81)*ln(5/(3+x))-x^3+20*x^2-118*x+162)/(x^2-18*x+8 
1))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-2187)*ln(5/(3+ 
x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x,method=_RETURNVERBOSE)
 

Output:

ln(((-x^2+18*x-81)*ln(5/(3+x))-x^3+20*x^2-118*x+162)/(x^2-18*x+81))*x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=x \log \left (-\frac {x^{3} - 20 \, x^{2} + {\left (x^{2} - 18 \, x + 81\right )} \log \left (\frac {5}{x + 3}\right ) + 118 \, x - 162}{x^{2} - 18 \, x + 81}\right ) \] Input:

integrate((((x^4-24*x^3+162*x^2-2187)*log(5/(3+x))+x^5-26*x^4+211*x^3-330* 
x^2-2214*x+4374)*log(((-x^2+18*x-81)*log(5/(3+x))-x^3+20*x^2-118*x+162)/(x 
^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-2187) 
*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x, algorithm="fricas 
")
 

Output:

x*log(-(x^3 - 20*x^2 + (x^2 - 18*x + 81)*log(5/(x + 3)) + 118*x - 162)/(x^ 
2 - 18*x + 81))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (19) = 38\).

Time = 0.71 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=\left (x - 1\right ) \log {\left (\frac {- x^{3} + 20 x^{2} - 118 x + \left (- x^{2} + 18 x - 81\right ) \log {\left (\frac {5}{x + 3} \right )} + 162}{x^{2} - 18 x + 81} \right )} + \log {\left (\log {\left (\frac {5}{x + 3} \right )} + \frac {x^{3} - 20 x^{2} + 118 x - 162}{x^{2} - 18 x + 81} \right )} \] Input:

integrate((((x**4-24*x**3+162*x**2-2187)*ln(5/(3+x))+x**5-26*x**4+211*x**3 
-330*x**2-2214*x+4374)*ln(((-x**2+18*x-81)*ln(5/(3+x))-x**3+20*x**2-118*x+ 
162)/(x**2-18*x+81))+x**5-25*x**4+188*x**3-255*x**2-1485*x)/((x**4-24*x**3 
+162*x**2-2187)*ln(5/(3+x))+x**5-26*x**4+211*x**3-330*x**2-2214*x+4374),x)
 

Output:

(x - 1)*log((-x**3 + 20*x**2 - 118*x + (-x**2 + 18*x - 81)*log(5/(x + 3)) 
+ 162)/(x**2 - 18*x + 81)) + log(log(5/(x + 3)) + (x**3 - 20*x**2 + 118*x 
- 162)/(x**2 - 18*x + 81))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).

Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=x \log \left (-x^{3} - x^{2} {\left (\log \left (5\right ) - \log \left (x + 3\right ) - 20\right )} + 2 \, x {\left (9 \, \log \left (5\right ) - 9 \, \log \left (x + 3\right ) - 59\right )} - 81 \, \log \left (5\right ) + 81 \, \log \left (x + 3\right ) + 162\right ) - 2 \, x \log \left (x - 9\right ) \] Input:

integrate((((x^4-24*x^3+162*x^2-2187)*log(5/(3+x))+x^5-26*x^4+211*x^3-330* 
x^2-2214*x+4374)*log(((-x^2+18*x-81)*log(5/(3+x))-x^3+20*x^2-118*x+162)/(x 
^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-2187) 
*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x, algorithm="maxima 
")
 

Output:

x*log(-x^3 - x^2*(log(5) - log(x + 3) - 20) + 2*x*(9*log(5) - 9*log(x + 3) 
 - 59) - 81*log(5) + 81*log(x + 3) + 162) - 2*x*log(x - 9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).

Time = 0.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=x \log \left (-x^{3} - x^{2} \log \left (\frac {5}{x + 3}\right ) + 20 \, x^{2} + 18 \, x \log \left (\frac {5}{x + 3}\right ) - 118 \, x - 81 \, \log \left (\frac {5}{x + 3}\right ) + 162\right ) - x \log \left (x^{2} - 18 \, x + 81\right ) \] Input:

integrate((((x^4-24*x^3+162*x^2-2187)*log(5/(3+x))+x^5-26*x^4+211*x^3-330* 
x^2-2214*x+4374)*log(((-x^2+18*x-81)*log(5/(3+x))-x^3+20*x^2-118*x+162)/(x 
^2-18*x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-2187) 
*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x, algorithm="giac")
 

Output:

x*log(-x^3 - x^2*log(5/(x + 3)) + 20*x^2 + 18*x*log(5/(x + 3)) - 118*x - 8 
1*log(5/(x + 3)) + 162) - x*log(x^2 - 18*x + 81)
 

Mupad [B] (verification not implemented)

Time = 4.39 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=x\,\ln \left (-\frac {118\,x+\ln \left (\frac {5}{x+3}\right )\,\left (x^2-18\,x+81\right )-20\,x^2+x^3-162}{x^2-18\,x+81}\right ) \] Input:

int(-(1485*x - log(-(118*x + log(5/(x + 3))*(x^2 - 18*x + 81) - 20*x^2 + x 
^3 - 162)/(x^2 - 18*x + 81))*(log(5/(x + 3))*(162*x^2 - 24*x^3 + x^4 - 218 
7) - 2214*x - 330*x^2 + 211*x^3 - 26*x^4 + x^5 + 4374) + 255*x^2 - 188*x^3 
 + 25*x^4 - x^5)/(log(5/(x + 3))*(162*x^2 - 24*x^3 + x^4 - 2187) - 2214*x 
- 330*x^2 + 211*x^3 - 26*x^4 + x^5 + 4374),x)
 

Output:

x*log(-(118*x + log(5/(x + 3))*(x^2 - 18*x + 81) - 20*x^2 + x^3 - 162)/(x^ 
2 - 18*x + 81))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 182, normalized size of antiderivative = 6.50 \[ \int \frac {-1485 x-255 x^2+188 x^3-25 x^4+x^5+\left (4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )\right ) \log \left (\frac {162-118 x+20 x^2-x^3+\left (-81+18 x-x^2\right ) \log \left (\frac {5}{3+x}\right )}{81-18 x+x^2}\right )}{4374-2214 x-330 x^2+211 x^3-26 x^4+x^5+\left (-2187+162 x^2-24 x^3+x^4\right ) \log \left (\frac {5}{3+x}\right )} \, dx=-25 \,\mathrm {log}\left (\mathrm {log}\left (\frac {5}{x +3}\right ) x^{2}-18 \,\mathrm {log}\left (\frac {5}{x +3}\right ) x +81 \,\mathrm {log}\left (\frac {5}{x +3}\right )+x^{3}-20 x^{2}+118 x -162\right )+50 \,\mathrm {log}\left (x -9\right )+\mathrm {log}\left (\frac {-\mathrm {log}\left (\frac {5}{x +3}\right ) x^{2}+18 \,\mathrm {log}\left (\frac {5}{x +3}\right ) x -81 \,\mathrm {log}\left (\frac {5}{x +3}\right )-x^{3}+20 x^{2}-118 x +162}{x^{2}-18 x +81}\right ) x +25 \,\mathrm {log}\left (\frac {-\mathrm {log}\left (\frac {5}{x +3}\right ) x^{2}+18 \,\mathrm {log}\left (\frac {5}{x +3}\right ) x -81 \,\mathrm {log}\left (\frac {5}{x +3}\right )-x^{3}+20 x^{2}-118 x +162}{x^{2}-18 x +81}\right ) \] Input:

int((((x^4-24*x^3+162*x^2-2187)*log(5/(3+x))+x^5-26*x^4+211*x^3-330*x^2-22 
14*x+4374)*log(((-x^2+18*x-81)*log(5/(3+x))-x^3+20*x^2-118*x+162)/(x^2-18* 
x+81))+x^5-25*x^4+188*x^3-255*x^2-1485*x)/((x^4-24*x^3+162*x^2-2187)*log(5 
/(3+x))+x^5-26*x^4+211*x^3-330*x^2-2214*x+4374),x)
                                                                                    
                                                                                    
 

Output:

 - 25*log(log(5/(x + 3))*x**2 - 18*log(5/(x + 3))*x + 81*log(5/(x + 3)) + 
x**3 - 20*x**2 + 118*x - 162) + 50*log(x - 9) + log(( - log(5/(x + 3))*x** 
2 + 18*log(5/(x + 3))*x - 81*log(5/(x + 3)) - x**3 + 20*x**2 - 118*x + 162 
)/(x**2 - 18*x + 81))*x + 25*log(( - log(5/(x + 3))*x**2 + 18*log(5/(x + 3 
))*x - 81*log(5/(x + 3)) - x**3 + 20*x**2 - 118*x + 162)/(x**2 - 18*x + 81 
))