\(\int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+(15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8) \log (x)+(-5-80 x^3) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+(-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9) \log (x)+(x^2+8 x^5+16 x^8) \log ^2(x)} \, dx\) [1403]

Optimal result

Integrand size = 180, antiderivative size = 28 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=\frac {-4+3 x+\frac {5 \log (x)}{x+4 x^4}}{-3-x+\log (x)} \] Output:

(3*x+5/(4*x^4+x)*ln(x)-4)/(ln(x)-3-x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {x \left (-4+3 x-16 x^3+12 x^4\right )+5 \log (x)}{x \left (1+4 x^3\right ) (3+x-\log (x))} \] Input:

Integrate[(-15 - x - 16*x^2 - 60*x^3 + 12*x^4 - 128*x^5 + 64*x^7 - 256*x^8 
 + (15 + 10*x + 3*x^2 + 240*x^3 + 100*x^4 + 24*x^5 + 48*x^8)*Log[x] + (-5 
- 80*x^3)*Log[x]^2)/(9*x^2 + 6*x^3 + x^4 + 72*x^5 + 48*x^6 + 8*x^7 + 144*x 
^8 + 96*x^9 + 16*x^10 + (-6*x^2 - 2*x^3 - 48*x^5 - 16*x^6 - 96*x^8 - 32*x^ 
9)*Log[x] + (x^2 + 8*x^5 + 16*x^8)*Log[x]^2),x]
 

Output:

-((x*(-4 + 3*x - 16*x^3 + 12*x^4) + 5*Log[x])/(x*(1 + 4*x^3)*(3 + x - Log[ 
x])))
 

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.

\(\displaystyle \int \frac {-256 x^8+64 x^7-128 x^5+12 x^4-60 x^3+\left (-80 x^3-5\right ) \log ^2(x)-16 x^2+\left (48 x^8+24 x^5+100 x^4+240 x^3+3 x^2+10 x+15\right ) \log (x)-x-15}{16 x^{10}+96 x^9+144 x^8+8 x^7+48 x^6+72 x^5+x^4+6 x^3+9 x^2+\left (16 x^8+8 x^5+x^2\right ) \log ^2(x)+\left (-32 x^9-96 x^8-16 x^6-48 x^5-2 x^3-6 x^2\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-256 x^8+64 x^7-128 x^5+12 x^4-60 x^3-\left (80 x^3+5\right ) \log ^2(x)-16 x^2+\left (48 x^8+24 x^5+100 x^4+240 x^3+3 x^2+10 x+15\right ) \log (x)-x-15}{x^2 \left (4 x^3+1\right )^2 (x-\log (x)+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {5 \left (16 x^3+1\right )}{x^2 \left (4 x^3+1\right )^2}-\frac {3 \left (16 x^8+8 x^5-20 x^4-80 x^3+x^2-5\right )}{x^2 \left (4 x^3+1\right )^2 (x-\log (x)+3)}+\frac {12 x^6-28 x^5+16 x^4+3 x^3-2 x^2+14 x-15}{x^2 \left (4 x^3+1\right ) (x-\log (x)+3)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 180 \int \frac {x}{\left (4 x^3+1\right )^2 (x-\log (x)+3)}dx-15 \int \frac {1}{x^2 (x-\log (x)+3)^2}dx+15 \int \frac {1}{x^2 (x-\log (x)+3)}dx+60 \int \frac {x^2}{\left (4 x^3+1\right )^2 (x-\log (x)+3)}dx-7 \int \frac {1}{(x-\log (x)+3)^2}dx+14 \int \frac {1}{x (x-\log (x)+3)^2}dx+3 \int \frac {x}{(x-\log (x)+3)^2}dx-\frac {5}{3} \int \frac {1}{\left (-(-2)^{2/3} x-1\right ) (x-\log (x)+3)^2}dx+10 \sqrt [3]{-2} \int \frac {1}{\left ((-2)^{2/3} x+1\right ) (x-\log (x)+3)^2}dx-\frac {5}{3} \int \frac {1}{\left (-2^{2/3} x-1\right ) (x-\log (x)+3)^2}dx-\frac {10}{3} 2^{2/3} \int \frac {1}{\left (2^{2/3} x+1\right ) (x-\log (x)+3)^2}dx-10 \sqrt [3]{2} \int \frac {1}{\left (2^{2/3} x+1\right ) (x-\log (x)+3)^2}dx-\frac {10}{3} 2^{2/3} \int \frac {1}{\left (2^{2/3} x-\sqrt [3]{-1}\right ) (x-\log (x)+3)^2}dx-\frac {10}{3} 2^{2/3} \int \frac {1}{\left (2^{2/3} x+(-1)^{2/3}\right ) (x-\log (x)+3)^2}dx-10 (-1)^{2/3} \sqrt [3]{2} \int \frac {1}{\left (1-\sqrt [3]{-1} 2^{2/3} x\right ) (x-\log (x)+3)^2}dx-\frac {5}{3} \int \frac {1}{\left (\sqrt [3]{-1} 2^{2/3} x-1\right ) (x-\log (x)+3)^2}dx-3 \int \frac {1}{x-\log (x)+3}dx-10 \sqrt [3]{-2} \int \frac {1}{\left ((-2)^{2/3} x+1\right ) (x-\log (x)+3)}dx+10 \sqrt [3]{2} \int \frac {1}{\left (2^{2/3} x+1\right ) (x-\log (x)+3)}dx+10 (-1)^{2/3} \sqrt [3]{2} \int \frac {1}{\left (1-\sqrt [3]{-1} 2^{2/3} x\right ) (x-\log (x)+3)}dx+\frac {5}{x \left (4 x^3+1\right )}\)

Input:

Int[(-15 - x - 16*x^2 - 60*x^3 + 12*x^4 - 128*x^5 + 64*x^7 - 256*x^8 + (15 
 + 10*x + 3*x^2 + 240*x^3 + 100*x^4 + 24*x^5 + 48*x^8)*Log[x] + (-5 - 80*x 
^3)*Log[x]^2)/(9*x^2 + 6*x^3 + x^4 + 72*x^5 + 48*x^6 + 8*x^7 + 144*x^8 + 9 
6*x^9 + 16*x^10 + (-6*x^2 - 2*x^3 - 48*x^5 - 16*x^6 - 96*x^8 - 32*x^9)*Log 
[x] + (x^2 + 8*x^5 + 16*x^8)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96

method	result	size
parallelrisch	\(\frac {16 x -48 x^{5}-20 \ln \left (x \right )-12 x^{2}+64 x^{4}}{4 x \left (4 x^{4}-4 x^{3} \ln \left (x \right )+12 x^{3}+x -\ln \left (x \right )+3\right )}\)	\(55\)
risch	\(\frac {5}{\left (4 x^{3}+1\right ) x}-\frac {12 x^{5}-16 x^{4}+3 x^{2}+x +15}{\left (4 x^{3}+1\right ) x \left (-\ln \left (x \right )+3+x \right )}\)	\(57\)
default	\(\frac {-88 x^{3} \ln \left (x \right )-22 \ln \left (x \right )+\frac {5 \ln \left (x \right )^{2}}{x}+12 x^{3} \ln \left (x \right )^{2}-\frac {15 \ln \left (x \right )}{x}+3 \ln \left (x \right )^{2}+156 x^{3}+39}{\left (\ln \left (x \right )-3\right ) \left (4 x^{3} \ln \left (x \right )-4 x^{4}-12 x^{3}+\ln \left (x \right )-x -3\right )}\)	\(83\)

Input:

int(((-80*x^3-5)*ln(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+15)*ln( 
x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^5+x^2)*l 
n(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*ln(x)+16*x^10+96*x^9+144 
*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x,method=_RETURNVERBOSE)
 

Output:

1/4/x*(16*x-48*x^5-20*ln(x)-12*x^2+64*x^4)/(4*x^4-4*x^3*ln(x)+12*x^3+x-ln( 
x)+3)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {12 \, x^{5} - 16 \, x^{4} + 3 \, x^{2} - 4 \, x + 5 \, \log \left (x\right )}{4 \, x^{5} + 12 \, x^{4} + x^{2} - {\left (4 \, x^{4} + x\right )} \log \left (x\right ) + 3 \, x} \] Input:

integrate(((-80*x^3-5)*log(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+ 
15)*log(x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^ 
5+x^2)*log(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*log(x)+16*x^10+ 
96*x^9+144*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x, algorithm="fricas")
 

Output:

-(12*x^5 - 16*x^4 + 3*x^2 - 4*x + 5*log(x))/(4*x^5 + 12*x^4 + x^2 - (4*x^4 
 + x)*log(x) + 3*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=\frac {12 x^{5} - 16 x^{4} + 3 x^{2} + x + 15}{- 4 x^{5} - 12 x^{4} - x^{2} - 3 x + \left (4 x^{4} + x\right ) \log {\left (x \right )}} + \frac {5}{4 x^{4} + x} \] Input:

integrate(((-80*x**3-5)*ln(x)**2+(48*x**8+24*x**5+100*x**4+240*x**3+3*x**2 
+10*x+15)*ln(x)-256*x**8+64*x**7-128*x**5+12*x**4-60*x**3-16*x**2-x-15)/(( 
16*x**8+8*x**5+x**2)*ln(x)**2+(-32*x**9-96*x**8-16*x**6-48*x**5-2*x**3-6*x 
**2)*ln(x)+16*x**10+96*x**9+144*x**8+8*x**7+48*x**6+72*x**5+x**4+6*x**3+9* 
x**2),x)
 

Output:

(12*x**5 - 16*x**4 + 3*x**2 + x + 15)/(-4*x**5 - 12*x**4 - x**2 - 3*x + (4 
*x**4 + x)*log(x)) + 5/(4*x**4 + x)
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {12 \, x^{5} - 16 \, x^{4} + 3 \, x^{2} - 4 \, x + 5 \, \log \left (x\right )}{4 \, x^{5} + 12 \, x^{4} + x^{2} - {\left (4 \, x^{4} + x\right )} \log \left (x\right ) + 3 \, x} \] Input:

integrate(((-80*x^3-5)*log(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+ 
15)*log(x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^ 
5+x^2)*log(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*log(x)+16*x^10+ 
96*x^9+144*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x, algorithm="maxima")
 

Output:

-(12*x^5 - 16*x^4 + 3*x^2 - 4*x + 5*log(x))/(4*x^5 + 12*x^4 + x^2 - (4*x^4 
 + x)*log(x) + 3*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).

Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {20 \, x^{2}}{4 \, x^{3} + 1} - \frac {12 \, x^{5} - 16 \, x^{4} + 3 \, x^{2} + x + 15}{4 \, x^{5} - 4 \, x^{4} \log \left (x\right ) + 12 \, x^{4} + x^{2} - x \log \left (x\right ) + 3 \, x} + \frac {5}{x} \] Input:

integrate(((-80*x^3-5)*log(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+ 
15)*log(x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^ 
5+x^2)*log(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*log(x)+16*x^10+ 
96*x^9+144*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x, algorithm="giac")
 

Output:

-20*x^2/(4*x^3 + 1) - (12*x^5 - 16*x^4 + 3*x^2 + x + 15)/(4*x^5 - 4*x^4*lo 
g(x) + 12*x^4 + x^2 - x*log(x) + 3*x) + 5/x
 

Mupad [B] (verification not implemented)

Time = 2.97 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {5\,\ln \left (x\right )+x\,\left (3\,\ln \left (x\right )-13\right )+x^4\,\left (12\,\ln \left (x\right )-52\right )}{x\,\left (4\,x^3+1\right )\,\left (x-\ln \left (x\right )+3\right )} \] Input:

int(-(x + log(x)^2*(80*x^3 + 5) - log(x)*(10*x + 3*x^2 + 240*x^3 + 100*x^4 
 + 24*x^5 + 48*x^8 + 15) + 16*x^2 + 60*x^3 - 12*x^4 + 128*x^5 - 64*x^7 + 2 
56*x^8 + 15)/(log(x)^2*(x^2 + 8*x^5 + 16*x^8) + 9*x^2 + 6*x^3 + x^4 + 72*x 
^5 + 48*x^6 + 8*x^7 + 144*x^8 + 96*x^9 + 16*x^10 - log(x)*(6*x^2 + 2*x^3 + 
 48*x^5 + 16*x^6 + 96*x^8 + 32*x^9)),x)
 

Output:

-(5*log(x) + x*(3*log(x) - 13) + x^4*(12*log(x) - 52))/(x*(4*x^3 + 1)*(x - 
 log(x) + 3))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=\frac {12 \,\mathrm {log}\left (x \right ) x^{4}+3 \,\mathrm {log}\left (x \right ) x +5 \,\mathrm {log}\left (x \right )-52 x^{4}-13 x}{x \left (4 \,\mathrm {log}\left (x \right ) x^{3}+\mathrm {log}\left (x \right )-4 x^{4}-12 x^{3}-x -3\right )} \] Input:

int(((-80*x^3-5)*log(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+15)*lo 
g(x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^5+x^2) 
*log(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*log(x)+16*x^10+96*x^9 
+144*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x)
 

Output:

(12*log(x)*x**4 + 3*log(x)*x + 5*log(x) - 52*x**4 - 13*x)/(x*(4*log(x)*x** 
3 + log(x) - 4*x**4 - 12*x**3 - x - 3))