Integrand size = 169, antiderivative size = 31 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\frac {4 \log ^2(x)}{\left (\frac {3}{x (4+4 x)}-\log \left (\left (2+e^5\right ) x\right )\right )^2} \] Output:
4*ln(x)^2/(3/(4+4*x)/x-ln(x*(exp(5)+2)))^2
Timed out. \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\text {\$Aborted} \] Input:
Integrate[((-384*x - 768*x^2 - 384*x^3)*Log[x] + (-384*x - 1664*x^2 - 2304 *x^3 - 1536*x^4 - 512*x^5)*Log[x]^2 + (512*x^2 + 1536*x^3 + 1536*x^4 + 512 *x^5)*Log[x]*Log[2*x + E^5*x])/(-27 + (108*x + 108*x^2)*Log[2*x + E^5*x] + (-144*x^2 - 288*x^3 - 144*x^4)*Log[2*x + E^5*x]^2 + (64*x^3 + 192*x^4 + 1 92*x^5 + 64*x^6)*Log[2*x + E^5*x]^3),x]
Output:
$Aborted
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.
| \(\displaystyle \int \frac {\left (-384 x^3-768 x^2-384 x\right ) \log (x)+\left (-512 x^5-1536 x^4-2304 x^3-1664 x^2-384 x\right ) \log ^2(x)+\left (512 x^5+1536 x^4+1536 x^3+512 x^2\right ) \log \left (e^5 x+2 x\right ) \log (x)}{\left (108 x^2+108 x\right ) \log \left (e^5 x+2 x\right )+\left (-144 x^4-288 x^3-144 x^2\right ) \log ^2\left (e^5 x+2 x\right )+\left (64 x^6+192 x^5+192 x^4+64 x^3\right ) \log ^3\left (e^5 x+2 x\right )-27} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {128 x (x+1) \log (x) \left (3 (2 x+1) \log (x)-(x+1) \left (4 x^2 \log \left (2+e^5\right )+4 x \log \left (2+e^5\right )-3\right )\right )}{\left (3-4 x (x+1) \log \left (\left (2+e^5\right ) x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 128 \int \frac {x (x+1) \log (x) \left ((x+1) \left (-4 \log \left (2+e^5\right ) x^2-4 \log \left (2+e^5\right ) x+3\right )+3 (2 x+1) \log (x)\right )}{\left (3-4 x (x+1) \log \left (\left (2+e^5\right ) x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 128 \int \left (\frac {\log (x) \left (-4 \log \left (2+e^5\right ) x^3-8 \log \left (2+e^5\right ) x^2+6 \log (x) x+3 \left (1-\frac {4}{3} \log \left (2+e^5\right )\right ) x+3 \log (x)+3\right ) x^2}{\left (-4 \log \left (\left (2+e^5\right ) x\right ) x^2-4 \log \left (\left (2+e^5\right ) x\right ) x+3\right )^3}+\frac {\log (x) \left (-4 \log \left (2+e^5\right ) x^3-8 \log \left (2+e^5\right ) x^2+6 \log (x) x+3 \left (1-\frac {4}{3} \log \left (2+e^5\right )\right ) x+3 \log (x)+3\right ) x}{\left (-4 \log \left (\left (2+e^5\right ) x\right ) x^2-4 \log \left (\left (2+e^5\right ) x\right ) x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 128 \left (-3 \int \frac {x \log ^2(x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx-9 \int \frac {x^2 \log ^2(x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx-3 \int \frac {x \log (x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx-\left (3-4 \log \left (2+e^5\right )\right ) \int \frac {x^2 \log (x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx-3 \int \frac {x^2 \log (x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx+4 \log \left (2+e^5\right ) \int \frac {x^5 \log (x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx+12 \log \left (2+e^5\right ) \int \frac {x^4 \log (x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx-6 \int \frac {x^3 \log ^2(x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx+8 \log \left (2+e^5\right ) \int \frac {x^3 \log (x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx-\left (3-4 \log \left (2+e^5\right )\right ) \int \frac {x^3 \log (x)}{\left (4 \log \left (\left (2+e^5\right ) x\right ) x^2+4 \log \left (\left (2+e^5\right ) x\right ) x-3\right )^3}dx\right )\) |
Input:
Int[((-384*x - 768*x^2 - 384*x^3)*Log[x] + (-384*x - 1664*x^2 - 2304*x^3 - 1536*x^4 - 512*x^5)*Log[x]^2 + (512*x^2 + 1536*x^3 + 1536*x^4 + 512*x^5)* Log[x]*Log[2*x + E^5*x])/(-27 + (108*x + 108*x^2)*Log[2*x + E^5*x] + (-144 *x^2 - 288*x^3 - 144*x^4)*Log[2*x + E^5*x]^2 + (64*x^3 + 192*x^4 + 192*x^5 + 64*x^6)*Log[2*x + E^5*x]^3),x]
Output:
$Aborted
Time = 4.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(\frac {96 x^{2} \ln \left (x \right )+96 x \ln \left (x \right )-36}{\left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )^{2}}\) | \(33\) |
| parallelrisch | \(\frac {18432 x^{4} \ln \left (x \right )^{2}+36864 x^{3} \ln \left (x \right )^{2}+18432 x^{2} \ln \left (x \right )^{2}}{4608 {\ln \left (x \left ({\mathrm e}^{5}+2\right )\right )}^{2} x^{4}+9216 {\ln \left (x \left ({\mathrm e}^{5}+2\right )\right )}^{2} x^{3}+4608 {\ln \left (x \left ({\mathrm e}^{5}+2\right )\right )}^{2} x^{2}-6912 \ln \left (x \left ({\mathrm e}^{5}+2\right )\right ) x^{2}-6912 \ln \left (x \left ({\mathrm e}^{5}+2\right )\right ) x +2592}\) | \(99\) |
| default | \(-\frac {128 \left (1+x \right ) x^{2} \left (x^{2}+2 x +1\right ) \ln \left (x \right )}{\left (4 x^{3}+8 x^{2}+10 x +3\right ) \left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )}+\frac {64 x^{2} \ln \left (x \right ) \left (1+x \right ) \left (12 x^{4} \ln \left (x \right )+36 x^{3} \ln \left (x \right )+42 x^{2} \ln \left (x \right )+21 x \ln \left (x \right )-6 x^{2}+3 \ln \left (x \right )-12 x -6\right )}{\left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )^{2} \left (4 x^{3}+8 x^{2}+10 x +3\right )}\) | \(136\) |
| parts | \(-\frac {128 \left (1+x \right ) x^{2} \left (x^{2}+2 x +1\right ) \ln \left (x \right )}{\left (4 x^{3}+8 x^{2}+10 x +3\right ) \left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )}+\frac {64 x^{2} \ln \left (x \right ) \left (1+x \right ) \left (12 x^{4} \ln \left (x \right )+36 x^{3} \ln \left (x \right )+42 x^{2} \ln \left (x \right )+21 x \ln \left (x \right )-6 x^{2}+3 \ln \left (x \right )-12 x -6\right )}{\left (4 x^{2} \ln \left (x \right )+4 x \ln \left (x \right )-3\right )^{2} \left (4 x^{3}+8 x^{2}+10 x +3\right )}\) | \(136\) |
Input:
int(((512*x^5+1536*x^4+1536*x^3+512*x^2)*ln(x)*ln(x*exp(5)+2*x)+(-512*x^5- 1536*x^4-2304*x^3-1664*x^2-384*x)*ln(x)^2+(-384*x^3-768*x^2-384*x)*ln(x))/ ((64*x^6+192*x^5+192*x^4+64*x^3)*ln(x*exp(5)+2*x)^3+(-144*x^4-288*x^3-144* x^2)*ln(x*exp(5)+2*x)^2+(108*x^2+108*x)*ln(x*exp(5)+2*x)-27),x,method=_RET URNVERBOSE)
Output:
12*(8*x^2*ln(x)+8*x*ln(x)-3)/(4*x^2*ln(x)+4*x*ln(x)-3)^2
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (28) = 56\).
Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.84 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=-\frac {4 \, {\left (16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (e^{5} + 2\right )^{2} - 24 \, {\left (x^{2} + x\right )} \log \left (x\right ) - 8 \, {\left (3 \, x^{2} - 4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (e^{5} + 2\right ) + 9\right )}}{16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right )^{2} + 16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (e^{5} + 2\right )^{2} - 24 \, {\left (x^{2} + x\right )} \log \left (x\right ) - 8 \, {\left (3 \, x^{2} - 4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (e^{5} + 2\right ) + 9} \] Input:
integrate(((512*x^5+1536*x^4+1536*x^3+512*x^2)*log(x)*log(x*exp(5)+2*x)+(- 512*x^5-1536*x^4-2304*x^3-1664*x^2-384*x)*log(x)^2+(-384*x^3-768*x^2-384*x )*log(x))/((64*x^6+192*x^5+192*x^4+64*x^3)*log(x*exp(5)+2*x)^3+(-144*x^4-2 88*x^3-144*x^2)*log(x*exp(5)+2*x)^2+(108*x^2+108*x)*log(x*exp(5)+2*x)-27), x, algorithm="fricas")
Output:
-4*(16*(x^4 + 2*x^3 + x^2)*log(e^5 + 2)^2 - 24*(x^2 + x)*log(x) - 8*(3*x^2 - 4*(x^4 + 2*x^3 + x^2)*log(x) + 3*x)*log(e^5 + 2) + 9)/(16*(x^4 + 2*x^3 + x^2)*log(x)^2 + 16*(x^4 + 2*x^3 + x^2)*log(e^5 + 2)^2 - 24*(x^2 + x)*log (x) - 8*(3*x^2 - 4*(x^4 + 2*x^3 + x^2)*log(x) + 3*x)*log(e^5 + 2) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (24) = 48\).
Time = 0.50 (sec) , antiderivative size = 243, normalized size of antiderivative = 7.84 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\frac {- 64 x^{4} \log {\left (2 + e^{5} \right )}^{2} - 128 x^{3} \log {\left (2 + e^{5} \right )}^{2} - 64 x^{2} \log {\left (2 + e^{5} \right )}^{2} + 96 x^{2} \log {\left (2 + e^{5} \right )} + 96 x \log {\left (2 + e^{5} \right )} + \left (- 128 x^{4} \log {\left (2 + e^{5} \right )} - 256 x^{3} \log {\left (2 + e^{5} \right )} - 128 x^{2} \log {\left (2 + e^{5} \right )} + 96 x^{2} + 96 x\right ) \log {\left (x \right )} - 36}{16 x^{4} \log {\left (2 + e^{5} \right )}^{2} + 32 x^{3} \log {\left (2 + e^{5} \right )}^{2} - 24 x^{2} \log {\left (2 + e^{5} \right )} + 16 x^{2} \log {\left (2 + e^{5} \right )}^{2} - 24 x \log {\left (2 + e^{5} \right )} + \left (16 x^{4} + 32 x^{3} + 16 x^{2}\right ) \log {\left (x \right )}^{2} + \left (32 x^{4} \log {\left (2 + e^{5} \right )} + 64 x^{3} \log {\left (2 + e^{5} \right )} - 24 x^{2} + 32 x^{2} \log {\left (2 + e^{5} \right )} - 24 x\right ) \log {\left (x \right )} + 9} \] Input:
integrate(((512*x**5+1536*x**4+1536*x**3+512*x**2)*ln(x)*ln(x*exp(5)+2*x)+ (-512*x**5-1536*x**4-2304*x**3-1664*x**2-384*x)*ln(x)**2+(-384*x**3-768*x* *2-384*x)*ln(x))/((64*x**6+192*x**5+192*x**4+64*x**3)*ln(x*exp(5)+2*x)**3+ (-144*x**4-288*x**3-144*x**2)*ln(x*exp(5)+2*x)**2+(108*x**2+108*x)*ln(x*ex p(5)+2*x)-27),x)
Output:
(-64*x**4*log(2 + exp(5))**2 - 128*x**3*log(2 + exp(5))**2 - 64*x**2*log(2 + exp(5))**2 + 96*x**2*log(2 + exp(5)) + 96*x*log(2 + exp(5)) + (-128*x** 4*log(2 + exp(5)) - 256*x**3*log(2 + exp(5)) - 128*x**2*log(2 + exp(5)) + 96*x**2 + 96*x)*log(x) - 36)/(16*x**4*log(2 + exp(5))**2 + 32*x**3*log(2 + exp(5))**2 - 24*x**2*log(2 + exp(5)) + 16*x**2*log(2 + exp(5))**2 - 24*x* log(2 + exp(5)) + (16*x**4 + 32*x**3 + 16*x**2)*log(x)**2 + (32*x**4*log(2 + exp(5)) + 64*x**3*log(2 + exp(5)) - 24*x**2 + 32*x**2*log(2 + exp(5)) - 24*x)*log(x) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (28) = 56\).
Time = 0.20 (sec) , antiderivative size = 216, normalized size of antiderivative = 6.97 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=-\frac {4 \, {\left (16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 8 \, {\left (2 \, \log \left (e^{5} + 2\right )^{2} - 3 \, \log \left (e^{5} + 2\right )\right )} x^{2} + 8 \, {\left (4 \, x^{4} \log \left (e^{5} + 2\right ) + 8 \, x^{3} \log \left (e^{5} + 2\right ) + x^{2} {\left (4 \, \log \left (e^{5} + 2\right ) - 3\right )} - 3 \, x\right )} \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9\right )}}{16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 8 \, {\left (2 \, \log \left (e^{5} + 2\right )^{2} - 3 \, \log \left (e^{5} + 2\right )\right )} x^{2} + 16 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right )^{2} + 8 \, {\left (4 \, x^{4} \log \left (e^{5} + 2\right ) + 8 \, x^{3} \log \left (e^{5} + 2\right ) + x^{2} {\left (4 \, \log \left (e^{5} + 2\right ) - 3\right )} - 3 \, x\right )} \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9} \] Input:
integrate(((512*x^5+1536*x^4+1536*x^3+512*x^2)*log(x)*log(x*exp(5)+2*x)+(- 512*x^5-1536*x^4-2304*x^3-1664*x^2-384*x)*log(x)^2+(-384*x^3-768*x^2-384*x )*log(x))/((64*x^6+192*x^5+192*x^4+64*x^3)*log(x*exp(5)+2*x)^3+(-144*x^4-2 88*x^3-144*x^2)*log(x*exp(5)+2*x)^2+(108*x^2+108*x)*log(x*exp(5)+2*x)-27), x, algorithm="maxima")
Output:
-4*(16*x^4*log(e^5 + 2)^2 + 32*x^3*log(e^5 + 2)^2 + 8*(2*log(e^5 + 2)^2 - 3*log(e^5 + 2))*x^2 + 8*(4*x^4*log(e^5 + 2) + 8*x^3*log(e^5 + 2) + x^2*(4* log(e^5 + 2) - 3) - 3*x)*log(x) - 24*x*log(e^5 + 2) + 9)/(16*x^4*log(e^5 + 2)^2 + 32*x^3*log(e^5 + 2)^2 + 8*(2*log(e^5 + 2)^2 - 3*log(e^5 + 2))*x^2 + 16*(x^4 + 2*x^3 + x^2)*log(x)^2 + 8*(4*x^4*log(e^5 + 2) + 8*x^3*log(e^5 + 2) + x^2*(4*log(e^5 + 2) - 3) - 3*x)*log(x) - 24*x*log(e^5 + 2) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (28) = 56\).
Time = 0.16 (sec) , antiderivative size = 239, normalized size of antiderivative = 7.71 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=-\frac {4 \, {\left (32 \, x^{4} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 64 \, x^{3} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{2} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{2} \log \left (e^{5} + 2\right )^{2} - 24 \, x^{2} \log \left (x\right ) - 24 \, x^{2} \log \left (e^{5} + 2\right ) - 24 \, x \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9\right )}}{16 \, x^{4} \log \left (x\right )^{2} + 32 \, x^{4} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{4} \log \left (e^{5} + 2\right )^{2} + 32 \, x^{3} \log \left (x\right )^{2} + 64 \, x^{3} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 32 \, x^{3} \log \left (e^{5} + 2\right )^{2} + 16 \, x^{2} \log \left (x\right )^{2} + 32 \, x^{2} \log \left (x\right ) \log \left (e^{5} + 2\right ) + 16 \, x^{2} \log \left (e^{5} + 2\right )^{2} - 24 \, x^{2} \log \left (x\right ) - 24 \, x^{2} \log \left (e^{5} + 2\right ) - 24 \, x \log \left (x\right ) - 24 \, x \log \left (e^{5} + 2\right ) + 9} \] Input:
integrate(((512*x^5+1536*x^4+1536*x^3+512*x^2)*log(x)*log(x*exp(5)+2*x)+(- 512*x^5-1536*x^4-2304*x^3-1664*x^2-384*x)*log(x)^2+(-384*x^3-768*x^2-384*x )*log(x))/((64*x^6+192*x^5+192*x^4+64*x^3)*log(x*exp(5)+2*x)^3+(-144*x^4-2 88*x^3-144*x^2)*log(x*exp(5)+2*x)^2+(108*x^2+108*x)*log(x*exp(5)+2*x)-27), x, algorithm="giac")
Output:
-4*(32*x^4*log(x)*log(e^5 + 2) + 16*x^4*log(e^5 + 2)^2 + 64*x^3*log(x)*log (e^5 + 2) + 32*x^3*log(e^5 + 2)^2 + 32*x^2*log(x)*log(e^5 + 2) + 16*x^2*lo g(e^5 + 2)^2 - 24*x^2*log(x) - 24*x^2*log(e^5 + 2) - 24*x*log(x) - 24*x*lo g(e^5 + 2) + 9)/(16*x^4*log(x)^2 + 32*x^4*log(x)*log(e^5 + 2) + 16*x^4*log (e^5 + 2)^2 + 32*x^3*log(x)^2 + 64*x^3*log(x)*log(e^5 + 2) + 32*x^3*log(e^ 5 + 2)^2 + 16*x^2*log(x)^2 + 32*x^2*log(x)*log(e^5 + 2) + 16*x^2*log(e^5 + 2)^2 - 24*x^2*log(x) - 24*x^2*log(e^5 + 2) - 24*x*log(x) - 24*x*log(e^5 + 2) + 9)
Timed out. \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\int \frac {{\ln \left (x\right )}^2\,\left (512\,x^5+1536\,x^4+2304\,x^3+1664\,x^2+384\,x\right )+\ln \left (x\right )\,\left (384\,x^3+768\,x^2+384\,x\right )-\ln \left (2\,x+x\,{\mathrm {e}}^5\right )\,\ln \left (x\right )\,\left (512\,x^5+1536\,x^4+1536\,x^3+512\,x^2\right )}{\left (-64\,x^6-192\,x^5-192\,x^4-64\,x^3\right )\,{\ln \left (2\,x+x\,{\mathrm {e}}^5\right )}^3+\left (144\,x^4+288\,x^3+144\,x^2\right )\,{\ln \left (2\,x+x\,{\mathrm {e}}^5\right )}^2+\left (-108\,x^2-108\,x\right )\,\ln \left (2\,x+x\,{\mathrm {e}}^5\right )+27} \,d x \] Input:
int((log(x)^2*(384*x + 1664*x^2 + 2304*x^3 + 1536*x^4 + 512*x^5) + log(x)* (384*x + 768*x^2 + 384*x^3) - log(2*x + x*exp(5))*log(x)*(512*x^2 + 1536*x ^3 + 1536*x^4 + 512*x^5))/(log(2*x + x*exp(5))^2*(144*x^2 + 288*x^3 + 144* x^4) - log(2*x + x*exp(5))*(108*x + 108*x^2) - log(2*x + x*exp(5))^3*(64*x ^3 + 192*x^4 + 192*x^5 + 64*x^6) + 27),x)
Output:
int((log(x)^2*(384*x + 1664*x^2 + 2304*x^3 + 1536*x^4 + 512*x^5) + log(x)* (384*x + 768*x^2 + 384*x^3) - log(2*x + x*exp(5))*log(x)*(512*x^2 + 1536*x ^3 + 1536*x^4 + 512*x^5))/(log(2*x + x*exp(5))^2*(144*x^2 + 288*x^3 + 144* x^4) - log(2*x + x*exp(5))*(108*x + 108*x^2) - log(2*x + x*exp(5))^3*(64*x ^3 + 192*x^4 + 192*x^5 + 64*x^6) + 27), x)
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.23 \[ \int \frac {\left (-384 x-768 x^2-384 x^3\right ) \log (x)+\left (-384 x-1664 x^2-2304 x^3-1536 x^4-512 x^5\right ) \log ^2(x)+\left (512 x^2+1536 x^3+1536 x^4+512 x^5\right ) \log (x) \log \left (2 x+e^5 x\right )}{-27+\left (108 x+108 x^2\right ) \log \left (2 x+e^5 x\right )+\left (-144 x^2-288 x^3-144 x^4\right ) \log ^2\left (2 x+e^5 x\right )+\left (64 x^3+192 x^4+192 x^5+64 x^6\right ) \log ^3\left (2 x+e^5 x\right )} \, dx=\frac {64 \mathrm {log}\left (x \right )^{2} x^{2} \left (x^{2}+2 x +1\right )}{16 \mathrm {log}\left (e^{5} x +2 x \right )^{2} x^{4}+32 \mathrm {log}\left (e^{5} x +2 x \right )^{2} x^{3}+16 \mathrm {log}\left (e^{5} x +2 x \right )^{2} x^{2}-24 \,\mathrm {log}\left (e^{5} x +2 x \right ) x^{2}-24 \,\mathrm {log}\left (e^{5} x +2 x \right ) x +9} \] Input:
int(((512*x^5+1536*x^4+1536*x^3+512*x^2)*log(x)*log(x*exp(5)+2*x)+(-512*x^ 5-1536*x^4-2304*x^3-1664*x^2-384*x)*log(x)^2+(-384*x^3-768*x^2-384*x)*log( x))/((64*x^6+192*x^5+192*x^4+64*x^3)*log(x*exp(5)+2*x)^3+(-144*x^4-288*x^3 -144*x^2)*log(x*exp(5)+2*x)^2+(108*x^2+108*x)*log(x*exp(5)+2*x)-27),x)
Output:
(64*log(x)**2*x**2*(x**2 + 2*x + 1))/(16*log(e**5*x + 2*x)**2*x**4 + 32*lo g(e**5*x + 2*x)**2*x**3 + 16*log(e**5*x + 2*x)**2*x**2 - 24*log(e**5*x + 2 *x)*x**2 - 24*log(e**5*x + 2*x)*x + 9)