\(\int \frac {10+22 x+4 x^2+5 x^5+x^6+(10+4 x+15 x^4+3 x^5) \log (x)+(15 x^3+3 x^4) \log ^2(x)+(5 x^2+x^3) \log ^3(x)+(10+2 x+(10+2 x+15 x^4+3 x^5) \log (x)+(30 x^3+6 x^4) \log ^2(x)+(15 x^2+3 x^3) \log ^3(x)) \log (5+x)+((15 x^3+3 x^4) \log ^2(x)+(15 x^2+3 x^3) \log ^3(x)) \log ^2(5+x)+(5 x^2+x^3) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+(15 x^5+3 x^6) \log (x)+(15 x^4+3 x^5) \log ^2(x)+(5 x^3+x^4) \log ^3(x)+((15 x^5+3 x^6) \log (x)+(30 x^4+6 x^5) \log ^2(x)+(15 x^3+3 x^4) \log ^3(x)) \log (5+x)+((15 x^4+3 x^5) \log ^2(x)+(15 x^3+3 x^4) \log ^3(x)) \log ^2(5+x)+(5 x^3+x^4) \log ^3(x) \log ^3(5+x)} \, dx\) [808]

Optimal result

Integrand size = 354, antiderivative size = 22 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=\log (x)-\frac {1}{\left (x^2+\log (x) (x+x \log (5+x))\right )^2} \] Output:

ln(x)-1/(ln(x)*(x*ln(5+x)+x)+x^2)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=\log (x)-\frac {1}{x^2 (x+\log (x)+\log (x) \log (5+x))^2} \] Input:

Integrate[(10 + 22*x + 4*x^2 + 5*x^5 + x^6 + (10 + 4*x + 15*x^4 + 3*x^5)*L 
og[x] + (15*x^3 + 3*x^4)*Log[x]^2 + (5*x^2 + x^3)*Log[x]^3 + (10 + 2*x + ( 
10 + 2*x + 15*x^4 + 3*x^5)*Log[x] + (30*x^3 + 6*x^4)*Log[x]^2 + (15*x^2 + 
3*x^3)*Log[x]^3)*Log[5 + x] + ((15*x^3 + 3*x^4)*Log[x]^2 + (15*x^2 + 3*x^3 
)*Log[x]^3)*Log[5 + x]^2 + (5*x^2 + x^3)*Log[x]^3*Log[5 + x]^3)/(5*x^6 + x 
^7 + (15*x^5 + 3*x^6)*Log[x] + (15*x^4 + 3*x^5)*Log[x]^2 + (5*x^3 + x^4)*L 
og[x]^3 + ((15*x^5 + 3*x^6)*Log[x] + (30*x^4 + 6*x^5)*Log[x]^2 + (15*x^3 + 
 3*x^4)*Log[x]^3)*Log[5 + x] + ((15*x^4 + 3*x^5)*Log[x]^2 + (15*x^3 + 3*x^ 
4)*Log[x]^3)*Log[5 + x]^2 + (5*x^3 + x^4)*Log[x]^3*Log[5 + x]^3),x]
 

Output:

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

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.

\(\displaystyle \int \frac {x^6+5 x^5+4 x^2+\left (3 x^5+15 x^4+4 x+10\right ) \log (x)+\left (3 x^4+15 x^3\right ) \log ^2(x)+\left (x^3+5 x^2\right ) \log ^3(x)+\left (x^3+5 x^2\right ) \log ^3(x) \log ^3(x+5)+\left (\left (3 x^4+15 x^3\right ) \log ^2(x)+\left (3 x^3+15 x^2\right ) \log ^3(x)\right ) \log ^2(x+5)+\left (\left (3 x^5+15 x^4+2 x+10\right ) \log (x)+\left (6 x^4+30 x^3\right ) \log ^2(x)+\left (3 x^3+15 x^2\right ) \log ^3(x)+2 x+10\right ) \log (x+5)+22 x+10}{x^7+5 x^6+\left (3 x^6+15 x^5\right ) \log (x)+\left (3 x^5+15 x^4\right ) \log ^2(x)+\left (x^4+5 x^3\right ) \log ^3(x)+\left (x^4+5 x^3\right ) \log ^3(x) \log ^3(x+5)+\left (\left (3 x^5+15 x^4\right ) \log ^2(x)+\left (3 x^4+15 x^3\right ) \log ^3(x)\right ) \log ^2(x+5)+\left (\left (3 x^6+15 x^5\right ) \log (x)+\left (6 x^5+30 x^4\right ) \log ^2(x)+\left (3 x^4+15 x^3\right ) \log ^3(x)\right ) \log (x+5)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {(x+5) \left (x^5+4 x+2 \log (x+5)+2\right )+3 (x+5) x^3 \log ^2(x) (\log (x+5)+1)^2+(x+5) x^2 \log ^3(x) (\log (x+5)+1)^3+\log (x) \left (3 x^5+15 x^4+\left (3 x^5+15 x^4+2 x+10\right ) \log (x+5)+4 x+10\right )}{x^3 (x+5) (x+\log (x) (\log (x+5)+1))^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (\log (x)+1)}{x^3 \log (x) (x+\log (x)+\log (x) \log (x+5))^2}+\frac {2 \left (-x+\log ^2(x)+x \log (x)+5 \log (x)-5\right )}{x^2 (x+5) \log (x) (x+\log (x)+\log (x) \log (x+5))^3}+\frac {1}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {1}{x^3 (x+\log (x)+\log (x) \log (x+5))^2}dx+2 \int \frac {1}{x^3 \log (x) (x+\log (x)+\log (x) \log (x+5))^2}dx+2 \int \frac {1}{x^2 (x+\log (x)+\log (x) \log (x+5))^3}dx-2 \int \frac {1}{x^2 \log (x) (x+\log (x)+\log (x) \log (x+5))^3}dx+\frac {2}{5} \int \frac {\log (x)}{x^2 (x+\log (x)+\log (x) \log (x+5))^3}dx-\frac {2}{25} \int \frac {\log (x)}{x (x+\log (x)+\log (x) \log (x+5))^3}dx+\frac {2}{25} \int \frac {\log (x)}{(x+5) (x+\log (x)+\log (x) \log (x+5))^3}dx+\log (x)\)

Input:

Int[(10 + 22*x + 4*x^2 + 5*x^5 + x^6 + (10 + 4*x + 15*x^4 + 3*x^5)*Log[x] 
+ (15*x^3 + 3*x^4)*Log[x]^2 + (5*x^2 + x^3)*Log[x]^3 + (10 + 2*x + (10 + 2 
*x + 15*x^4 + 3*x^5)*Log[x] + (30*x^3 + 6*x^4)*Log[x]^2 + (15*x^2 + 3*x^3) 
*Log[x]^3)*Log[5 + x] + ((15*x^3 + 3*x^4)*Log[x]^2 + (15*x^2 + 3*x^3)*Log[ 
x]^3)*Log[5 + x]^2 + (5*x^2 + x^3)*Log[x]^3*Log[5 + x]^3)/(5*x^6 + x^7 + ( 
15*x^5 + 3*x^6)*Log[x] + (15*x^4 + 3*x^5)*Log[x]^2 + (5*x^3 + x^4)*Log[x]^ 
3 + ((15*x^5 + 3*x^6)*Log[x] + (30*x^4 + 6*x^5)*Log[x]^2 + (15*x^3 + 3*x^4 
)*Log[x]^3)*Log[5 + x] + ((15*x^4 + 3*x^5)*Log[x]^2 + (15*x^3 + 3*x^4)*Log 
[x]^3)*Log[5 + x]^2 + (5*x^3 + x^4)*Log[x]^3*Log[5 + x]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\ln \left (x \right )-\frac {1}{x^{2} \left (\ln \left (5+x \right ) \ln \left (x \right )+\ln \left (x \right )+x \right )^{2}}\]

Input:

int(((x^3+5*x^2)*ln(x)^3*ln(5+x)^3+((3*x^3+15*x^2)*ln(x)^3+(3*x^4+15*x^3)* 
ln(x)^2)*ln(5+x)^2+((3*x^3+15*x^2)*ln(x)^3+(6*x^4+30*x^3)*ln(x)^2+(3*x^5+1 
5*x^4+2*x+10)*ln(x)+2*x+10)*ln(5+x)+(x^3+5*x^2)*ln(x)^3+(3*x^4+15*x^3)*ln( 
x)^2+(3*x^5+15*x^4+4*x+10)*ln(x)+x^6+5*x^5+4*x^2+22*x+10)/((x^4+5*x^3)*ln( 
x)^3*ln(5+x)^3+((3*x^4+15*x^3)*ln(x)^3+(3*x^5+15*x^4)*ln(x)^2)*ln(5+x)^2+( 
(3*x^4+15*x^3)*ln(x)^3+(6*x^5+30*x^4)*ln(x)^2+(3*x^6+15*x^5)*ln(x))*ln(5+x 
)+(x^4+5*x^3)*ln(x)^3+(3*x^5+15*x^4)*ln(x)^2+(3*x^6+15*x^5)*ln(x)+x^7+5*x^ 
6),x)
 

Output:

ln(x)-1/x^2/(ln(5+x)*ln(x)+ln(x)+x)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (22) = 44\).

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 5.41 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=\frac {x^{2} \log \left (x + 5\right )^{2} \log \left (x\right )^{3} + x^{4} \log \left (x\right ) + 2 \, x^{3} \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{3} + 2 \, {\left (x^{3} \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{3}\right )} \log \left (x + 5\right ) - 1}{x^{2} \log \left (x + 5\right )^{2} \log \left (x\right )^{2} + x^{4} + 2 \, x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2}\right )} \log \left (x + 5\right )} \] Input:

integrate(((x^3+5*x^2)*log(x)^3*log(5+x)^3+((3*x^3+15*x^2)*log(x)^3+(3*x^4 
+15*x^3)*log(x)^2)*log(5+x)^2+((3*x^3+15*x^2)*log(x)^3+(6*x^4+30*x^3)*log( 
x)^2+(3*x^5+15*x^4+2*x+10)*log(x)+2*x+10)*log(5+x)+(x^3+5*x^2)*log(x)^3+(3 
*x^4+15*x^3)*log(x)^2+(3*x^5+15*x^4+4*x+10)*log(x)+x^6+5*x^5+4*x^2+22*x+10 
)/((x^4+5*x^3)*log(x)^3*log(5+x)^3+((3*x^4+15*x^3)*log(x)^3+(3*x^5+15*x^4) 
*log(x)^2)*log(5+x)^2+((3*x^4+15*x^3)*log(x)^3+(6*x^5+30*x^4)*log(x)^2+(3* 
x^6+15*x^5)*log(x))*log(5+x)+(x^4+5*x^3)*log(x)^3+(3*x^5+15*x^4)*log(x)^2+ 
(3*x^6+15*x^5)*log(x)+x^7+5*x^6),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).

Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=\log {\left (x \right )} - \frac {1}{x^{4} + 2 x^{3} \log {\left (x \right )} + x^{2} \log {\left (x \right )}^{2} \log {\left (x + 5 \right )}^{2} + x^{2} \log {\left (x \right )}^{2} + \left (2 x^{3} \log {\left (x \right )} + 2 x^{2} \log {\left (x \right )}^{2}\right ) \log {\left (x + 5 \right )}} \] Input:

integrate(((x**3+5*x**2)*ln(x)**3*ln(5+x)**3+((3*x**3+15*x**2)*ln(x)**3+(3 
*x**4+15*x**3)*ln(x)**2)*ln(5+x)**2+((3*x**3+15*x**2)*ln(x)**3+(6*x**4+30* 
x**3)*ln(x)**2+(3*x**5+15*x**4+2*x+10)*ln(x)+2*x+10)*ln(5+x)+(x**3+5*x**2) 
*ln(x)**3+(3*x**4+15*x**3)*ln(x)**2+(3*x**5+15*x**4+4*x+10)*ln(x)+x**6+5*x 
**5+4*x**2+22*x+10)/((x**4+5*x**3)*ln(x)**3*ln(5+x)**3+((3*x**4+15*x**3)*l 
n(x)**3+(3*x**5+15*x**4)*ln(x)**2)*ln(5+x)**2+((3*x**4+15*x**3)*ln(x)**3+( 
6*x**5+30*x**4)*ln(x)**2+(3*x**6+15*x**5)*ln(x))*ln(5+x)+(x**4+5*x**3)*ln( 
x)**3+(3*x**5+15*x**4)*ln(x)**2+(3*x**6+15*x**5)*ln(x)+x**7+5*x**6),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).

Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=-\frac {1}{x^{2} \log \left (x + 5\right )^{2} \log \left (x\right )^{2} + x^{4} + 2 \, x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2}\right )} \log \left (x + 5\right )} + \log \left (x\right ) \] Input:

integrate(((x^3+5*x^2)*log(x)^3*log(5+x)^3+((3*x^3+15*x^2)*log(x)^3+(3*x^4 
+15*x^3)*log(x)^2)*log(5+x)^2+((3*x^3+15*x^2)*log(x)^3+(6*x^4+30*x^3)*log( 
x)^2+(3*x^5+15*x^4+2*x+10)*log(x)+2*x+10)*log(5+x)+(x^3+5*x^2)*log(x)^3+(3 
*x^4+15*x^3)*log(x)^2+(3*x^5+15*x^4+4*x+10)*log(x)+x^6+5*x^5+4*x^2+22*x+10 
)/((x^4+5*x^3)*log(x)^3*log(5+x)^3+((3*x^4+15*x^3)*log(x)^3+(3*x^5+15*x^4) 
*log(x)^2)*log(5+x)^2+((3*x^4+15*x^3)*log(x)^3+(6*x^5+30*x^4)*log(x)^2+(3* 
x^6+15*x^5)*log(x))*log(5+x)+(x^4+5*x^3)*log(x)^3+(3*x^5+15*x^4)*log(x)^2+ 
(3*x^6+15*x^5)*log(x)+x^7+5*x^6),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (22) = 44\).

Time = 0.34 (sec) , antiderivative size = 281, normalized size of antiderivative = 12.77 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=-\frac {x \log \left (x\right ) + \log \left (x\right )^{2} - x + 5 \, \log \left (x\right ) - 5}{x^{3} \log \left (x + 5\right )^{2} \log \left (x\right )^{3} + x^{2} \log \left (x + 5\right )^{2} \log \left (x\right )^{4} + 2 \, x^{4} \log \left (x + 5\right ) \log \left (x\right )^{2} - x^{3} \log \left (x + 5\right )^{2} \log \left (x\right )^{2} + 4 \, x^{3} \log \left (x + 5\right ) \log \left (x\right )^{3} + 5 \, x^{2} \log \left (x + 5\right )^{2} \log \left (x\right )^{3} + 2 \, x^{2} \log \left (x + 5\right ) \log \left (x\right )^{4} + x^{5} \log \left (x\right ) - 2 \, x^{4} \log \left (x + 5\right ) \log \left (x\right ) + 3 \, x^{4} \log \left (x\right )^{2} + 8 \, x^{3} \log \left (x + 5\right ) \log \left (x\right )^{2} - 5 \, x^{2} \log \left (x + 5\right )^{2} \log \left (x\right )^{2} + 3 \, x^{3} \log \left (x\right )^{3} + 10 \, x^{2} \log \left (x + 5\right ) \log \left (x\right )^{3} + x^{2} \log \left (x\right )^{4} - x^{5} + 3 \, x^{4} \log \left (x\right ) - 10 \, x^{3} \log \left (x + 5\right ) \log \left (x\right ) + 9 \, x^{3} \log \left (x\right )^{2} - 10 \, x^{2} \log \left (x + 5\right ) \log \left (x\right )^{2} + 5 \, x^{2} \log \left (x\right )^{3} - 5 \, x^{4} - 10 \, x^{3} \log \left (x\right ) - 5 \, x^{2} \log \left (x\right )^{2}} + \log \left (x\right ) \] Input:

integrate(((x^3+5*x^2)*log(x)^3*log(5+x)^3+((3*x^3+15*x^2)*log(x)^3+(3*x^4 
+15*x^3)*log(x)^2)*log(5+x)^2+((3*x^3+15*x^2)*log(x)^3+(6*x^4+30*x^3)*log( 
x)^2+(3*x^5+15*x^4+2*x+10)*log(x)+2*x+10)*log(5+x)+(x^3+5*x^2)*log(x)^3+(3 
*x^4+15*x^3)*log(x)^2+(3*x^5+15*x^4+4*x+10)*log(x)+x^6+5*x^5+4*x^2+22*x+10 
)/((x^4+5*x^3)*log(x)^3*log(5+x)^3+((3*x^4+15*x^3)*log(x)^3+(3*x^5+15*x^4) 
*log(x)^2)*log(5+x)^2+((3*x^4+15*x^3)*log(x)^3+(6*x^5+30*x^4)*log(x)^2+(3* 
x^6+15*x^5)*log(x))*log(5+x)+(x^4+5*x^3)*log(x)^3+(3*x^5+15*x^4)*log(x)^2+ 
(3*x^6+15*x^5)*log(x)+x^7+5*x^6),x, algorithm="giac")
 

Output:

-(x*log(x) + log(x)^2 - x + 5*log(x) - 5)/(x^3*log(x + 5)^2*log(x)^3 + x^2 
*log(x + 5)^2*log(x)^4 + 2*x^4*log(x + 5)*log(x)^2 - x^3*log(x + 5)^2*log( 
x)^2 + 4*x^3*log(x + 5)*log(x)^3 + 5*x^2*log(x + 5)^2*log(x)^3 + 2*x^2*log 
(x + 5)*log(x)^4 + x^5*log(x) - 2*x^4*log(x + 5)*log(x) + 3*x^4*log(x)^2 + 
 8*x^3*log(x + 5)*log(x)^2 - 5*x^2*log(x + 5)^2*log(x)^2 + 3*x^3*log(x)^3 
+ 10*x^2*log(x + 5)*log(x)^3 + x^2*log(x)^4 - x^5 + 3*x^4*log(x) - 10*x^3* 
log(x + 5)*log(x) + 9*x^3*log(x)^2 - 10*x^2*log(x + 5)*log(x)^2 + 5*x^2*lo 
g(x)^3 - 5*x^4 - 10*x^3*log(x) - 5*x^2*log(x)^2) + log(x)
 

Mupad [B] (verification not implemented)

Time = 7.54 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.73 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=\frac {x^4\,\ln \left (x\right )+2\,x^3\,\ln \left (x+5\right )\,{\ln \left (x\right )}^2+2\,x^3\,{\ln \left (x\right )}^2+x^2\,{\ln \left (x+5\right )}^2\,{\ln \left (x\right )}^3+2\,x^2\,\ln \left (x+5\right )\,{\ln \left (x\right )}^3+x^2\,{\ln \left (x\right )}^3-1}{x^2\,{\left (x+\ln \left (x\right )+\ln \left (x+5\right )\,\ln \left (x\right )\right )}^2} \] Input:

int((22*x + log(x)^3*(5*x^2 + x^3) + log(x + 5)^2*(log(x)^3*(15*x^2 + 3*x^ 
3) + log(x)^2*(15*x^3 + 3*x^4)) + log(x)^2*(15*x^3 + 3*x^4) + log(x + 5)*( 
2*x + log(x)^3*(15*x^2 + 3*x^3) + log(x)^2*(30*x^3 + 6*x^4) + log(x)*(2*x 
+ 15*x^4 + 3*x^5 + 10) + 10) + 4*x^2 + 5*x^5 + x^6 + log(x)*(4*x + 15*x^4 
+ 3*x^5 + 10) + log(x + 5)^3*log(x)^3*(5*x^2 + x^3) + 10)/(log(x)*(15*x^5 
+ 3*x^6) + log(x)^3*(5*x^3 + x^4) + log(x + 5)^2*(log(x)^3*(15*x^3 + 3*x^4 
) + log(x)^2*(15*x^4 + 3*x^5)) + log(x)^2*(15*x^4 + 3*x^5) + 5*x^6 + x^7 + 
 log(x + 5)*(log(x)*(15*x^5 + 3*x^6) + log(x)^3*(15*x^3 + 3*x^4) + log(x)^ 
2*(30*x^4 + 6*x^5)) + log(x + 5)^3*log(x)^3*(5*x^3 + x^4)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 5.18 \[ \int \frac {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+x)} \, dx=\frac {\mathrm {log}\left (x +5\right )^{2} \mathrm {log}\left (x \right )^{3} x^{2}+2 \,\mathrm {log}\left (x +5\right ) \mathrm {log}\left (x \right )^{3} x^{2}+2 \,\mathrm {log}\left (x +5\right ) \mathrm {log}\left (x \right )^{2} x^{3}+\mathrm {log}\left (x \right )^{3} x^{2}+2 \mathrm {log}\left (x \right )^{2} x^{3}+\mathrm {log}\left (x \right ) x^{4}-1}{x^{2} \left (\mathrm {log}\left (x +5\right )^{2} \mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x +5\right ) \mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x +5\right ) \mathrm {log}\left (x \right ) x +\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) x +x^{2}\right )} \] Input:

int(((x^3+5*x^2)*log(x)^3*log(5+x)^3+((3*x^3+15*x^2)*log(x)^3+(3*x^4+15*x^ 
3)*log(x)^2)*log(5+x)^2+((3*x^3+15*x^2)*log(x)^3+(6*x^4+30*x^3)*log(x)^2+( 
3*x^5+15*x^4+2*x+10)*log(x)+2*x+10)*log(5+x)+(x^3+5*x^2)*log(x)^3+(3*x^4+1 
5*x^3)*log(x)^2+(3*x^5+15*x^4+4*x+10)*log(x)+x^6+5*x^5+4*x^2+22*x+10)/((x^ 
4+5*x^3)*log(x)^3*log(5+x)^3+((3*x^4+15*x^3)*log(x)^3+(3*x^5+15*x^4)*log(x 
)^2)*log(5+x)^2+((3*x^4+15*x^3)*log(x)^3+(6*x^5+30*x^4)*log(x)^2+(3*x^6+15 
*x^5)*log(x))*log(5+x)+(x^4+5*x^3)*log(x)^3+(3*x^5+15*x^4)*log(x)^2+(3*x^6 
+15*x^5)*log(x)+x^7+5*x^6),x)
 

Output:

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