Integrand size = 189, antiderivative size = 27 \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\frac {25}{\left (-\frac {x^2}{5 \left (-x+4 x^2\right )}+\log (1+x)\right )^4} \] Output:
25/(ln(1+x)-1/5*x^2/(4*x^2-x))^4
Time = 1.38 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\frac {15625 (1-4 x)^4}{(x+(5-20 x) \log (1+x))^4} \] Input:
Integrate[(375000 - 6937500*x + 52250000*x^2 - 201000000*x^3 + 396000000*x ^4 - 320000000*x^5)/(-x^5 - x^6 + (-25*x^4 + 75*x^5 + 100*x^6)*Log[1 + x] + (-250*x^3 + 1750*x^4 - 2000*x^5 - 4000*x^6)*Log[1 + x]^2 + (-1250*x^2 + 13750*x^3 - 45000*x^4 + 20000*x^5 + 80000*x^6)*Log[1 + x]^3 + (-3125*x + 4 6875*x^2 - 250000*x^3 + 500000*x^4 - 800000*x^6)*Log[1 + x]^4 + (-3125 + 5 9375*x - 437500*x^2 + 1500000*x^3 - 2000000*x^4 - 800000*x^5 + 3200000*x^6 )*Log[1 + x]^5),x]
Output:
(15625*(1 - 4*x)^4)/(x + (5 - 20*x)*Log[1 + x])^4
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.
| \(\displaystyle \int \frac {-320000000 x^5+396000000 x^4-201000000 x^3+52250000 x^2-6937500 x+375000}{-x^6-x^5+\left (100 x^6+75 x^5-25 x^4\right ) \log (x+1)+\left (-4000 x^6-2000 x^5+1750 x^4-250 x^3\right ) \log ^2(x+1)+\left (-800000 x^6+500000 x^4-250000 x^3+46875 x^2-3125 x\right ) \log ^4(x+1)+\left (3200000 x^6-800000 x^5-2000000 x^4+1500000 x^3-437500 x^2+59375 x-3125\right ) \log ^5(x+1)+\left (80000 x^6+20000 x^5-45000 x^4+13750 x^3-1250 x^2\right ) \log ^3(x+1)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {62500 (1-4 x)^3 \left (-80 x^2+39 x-6\right )}{(x+1) (x-5 (4 x-1) \log (x+1))^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 62500 \int -\frac {(1-4 x)^3 \left (80 x^2-39 x+6\right )}{(x+1) (x+5 (1-4 x) \log (x+1))^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -62500 \int \frac {(1-4 x)^3 \left (80 x^2-39 x+6\right )}{(x+1) (x+5 (1-4 x) \log (x+1))^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -62500 \int \left (\frac {5120 x^4}{(20 \log (x+1) x-x-5 \log (x+1))^5}-\frac {11456 x^3}{(20 \log (x+1) x-x-5 \log (x+1))^5}+\frac {14672 x^2}{(20 \log (x+1) x-x-5 \log (x+1))^5}-\frac {15508 x}{(20 \log (x+1) x-x-5 \log (x+1))^5}-\frac {15625}{(x+1) (20 \log (x+1) x-x-5 \log (x+1))^5}+\frac {15619}{(20 \log (x+1) x-x-5 \log (x+1))^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -62500 \left (5120 \int \frac {x^4}{(20 \log (x+1) x-x-5 \log (x+1))^5}dx-11456 \int \frac {x^3}{(20 \log (x+1) x-x-5 \log (x+1))^5}dx+14672 \int \frac {x^2}{(20 \log (x+1) x-x-5 \log (x+1))^5}dx+15619 \int \frac {1}{(20 \log (x+1) x-x-5 \log (x+1))^5}dx-15508 \int \frac {x}{(20 \log (x+1) x-x-5 \log (x+1))^5}dx-15625 \int \frac {1}{(x+1) (20 \log (x+1) x-x-5 \log (x+1))^5}dx\right )\) |
Input:
Int[(375000 - 6937500*x + 52250000*x^2 - 201000000*x^3 + 396000000*x^4 - 3 20000000*x^5)/(-x^5 - x^6 + (-25*x^4 + 75*x^5 + 100*x^6)*Log[1 + x] + (-25 0*x^3 + 1750*x^4 - 2000*x^5 - 4000*x^6)*Log[1 + x]^2 + (-1250*x^2 + 13750* x^3 - 45000*x^4 + 20000*x^5 + 80000*x^6)*Log[1 + x]^3 + (-3125*x + 46875*x ^2 - 250000*x^3 + 500000*x^4 - 800000*x^6)*Log[1 + x]^4 + (-3125 + 59375*x - 437500*x^2 + 1500000*x^3 - 2000000*x^4 - 800000*x^5 + 3200000*x^6)*Log[ 1 + x]^5),x]
Output:
$Aborted
Time = 0.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56
| method | result | size |
| risch | \(\frac {15625 \left (64 x^{3}-48 x^{2}+12 x -1\right ) \left (-1+4 x \right )}{\left (20 \ln \left (1+x \right ) x -5 \ln \left (1+x \right )-x \right )^{4}}\) | \(42\) |
| parallelrisch | \(\frac {640000000000 x^{4}-640000000000 x^{3}+240000000000 x^{2}-40000000000 x +2500000000}{25600000000 \ln \left (1+x \right )^{4} x^{4}-5120000000 \ln \left (1+x \right )^{3} x^{4}-25600000000 \ln \left (1+x \right )^{4} x^{3}+384000000 \ln \left (1+x \right )^{2} x^{4}+3840000000 \ln \left (1+x \right )^{3} x^{3}+9600000000 \ln \left (1+x \right )^{4} x^{2}-12800000 \ln \left (1+x \right ) x^{4}-192000000 \ln \left (1+x \right )^{2} x^{3}-960000000 \ln \left (1+x \right )^{3} x^{2}-1600000000 \ln \left (1+x \right )^{4} x +160000 x^{4}+3200000 x^{3} \ln \left (1+x \right )+24000000 x^{2} \ln \left (1+x \right )^{2}+80000000 x \ln \left (1+x \right )^{3}+100000000 \ln \left (1+x \right )^{4}}\) | \(172\) |
| derivativedivides | \(\frac {1600000000 \ln \left (1+x \right )^{2} \left (1+x \right )^{4}-160000000 \ln \left (1+x \right ) \left (1+x \right )^{4}-8000000000 \ln \left (1+x \right )^{2} \left (1+x \right )^{3}+4000000 \left (1+x \right )^{4}+800000000 \ln \left (1+x \right ) \left (1+x \right )^{3}+15000000000 \ln \left (1+x \right )^{2} \left (1+x \right )^{2}-20000000 \left (1+x \right )^{3}-1500000000 \ln \left (1+x \right ) \left (1+x \right )^{2}-12500000000 \left (1+x \right ) \ln \left (1+x \right )^{2}+37500000 \left (1+x \right )^{2}+1250000000 \ln \left (1+x \right ) \left (1+x \right )+3906250000 \ln \left (1+x \right )^{2}-21484375-31250000 x -390625000 \ln \left (1+x \right )}{\left (20 \ln \left (1+x \right ) \left (1+x \right )-x -25 \ln \left (1+x \right )\right )^{4} \left (400 \ln \left (1+x \right )^{2}-40 \ln \left (1+x \right )+1\right )}\) | \(174\) |
| default | \(\frac {1600000000 \ln \left (1+x \right )^{2} \left (1+x \right )^{4}-160000000 \ln \left (1+x \right ) \left (1+x \right )^{4}-8000000000 \ln \left (1+x \right )^{2} \left (1+x \right )^{3}+4000000 \left (1+x \right )^{4}+800000000 \ln \left (1+x \right ) \left (1+x \right )^{3}+15000000000 \ln \left (1+x \right )^{2} \left (1+x \right )^{2}-20000000 \left (1+x \right )^{3}-1500000000 \ln \left (1+x \right ) \left (1+x \right )^{2}-12500000000 \left (1+x \right ) \ln \left (1+x \right )^{2}+37500000 \left (1+x \right )^{2}+1250000000 \ln \left (1+x \right ) \left (1+x \right )+3906250000 \ln \left (1+x \right )^{2}-21484375-31250000 x -390625000 \ln \left (1+x \right )}{\left (20 \ln \left (1+x \right ) \left (1+x \right )-x -25 \ln \left (1+x \right )\right )^{4} \left (400 \ln \left (1+x \right )^{2}-40 \ln \left (1+x \right )+1\right )}\) | \(174\) |
Input:
int((-320000000*x^5+396000000*x^4-201000000*x^3+52250000*x^2-6937500*x+375 000)/((3200000*x^6-800000*x^5-2000000*x^4+1500000*x^3-437500*x^2+59375*x-3 125)*ln(1+x)^5+(-800000*x^6+500000*x^4-250000*x^3+46875*x^2-3125*x)*ln(1+x )^4+(80000*x^6+20000*x^5-45000*x^4+13750*x^3-1250*x^2)*ln(1+x)^3+(-4000*x^ 6-2000*x^5+1750*x^4-250*x^3)*ln(1+x)^2+(100*x^6+75*x^5-25*x^4)*ln(1+x)-x^6 -x^5),x,method=_RETURNVERBOSE)
Output:
15625*(64*x^3-48*x^2+12*x-1)*(-1+4*x)/(20*ln(1+x)*x-5*ln(1+x)-x)^4
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.52 \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\frac {15625 \, {\left (256 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 16 \, x + 1\right )}}{625 \, {\left (256 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 16 \, x + 1\right )} \log \left (x + 1\right )^{4} + x^{4} - 500 \, {\left (64 \, x^{4} - 48 \, x^{3} + 12 \, x^{2} - x\right )} \log \left (x + 1\right )^{3} + 150 \, {\left (16 \, x^{4} - 8 \, x^{3} + x^{2}\right )} \log \left (x + 1\right )^{2} - 20 \, {\left (4 \, x^{4} - x^{3}\right )} \log \left (x + 1\right )} \] Input:
integrate((-320000000*x^5+396000000*x^4-201000000*x^3+52250000*x^2-6937500 *x+375000)/((3200000*x^6-800000*x^5-2000000*x^4+1500000*x^3-437500*x^2+593 75*x-3125)*log(1+x)^5+(-800000*x^6+500000*x^4-250000*x^3+46875*x^2-3125*x) *log(1+x)^4+(80000*x^6+20000*x^5-45000*x^4+13750*x^3-1250*x^2)*log(1+x)^3+ (-4000*x^6-2000*x^5+1750*x^4-250*x^3)*log(1+x)^2+(100*x^6+75*x^5-25*x^4)*l og(1+x)-x^6-x^5),x, algorithm="fricas")
Output:
15625*(256*x^4 - 256*x^3 + 96*x^2 - 16*x + 1)/(625*(256*x^4 - 256*x^3 + 96 *x^2 - 16*x + 1)*log(x + 1)^4 + x^4 - 500*(64*x^4 - 48*x^3 + 12*x^2 - x)*l og(x + 1)^3 + 150*(16*x^4 - 8*x^3 + x^2)*log(x + 1)^2 - 20*(4*x^4 - x^3)*l og(x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (19) = 38\).
Time = 0.77 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.15 \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\frac {4000000 x^{4} - 4000000 x^{3} + 1500000 x^{2} - 250000 x + 15625}{x^{4} + \left (- 80 x^{4} + 20 x^{3}\right ) \log {\left (x + 1 \right )} + \left (2400 x^{4} - 1200 x^{3} + 150 x^{2}\right ) \log {\left (x + 1 \right )}^{2} + \left (- 32000 x^{4} + 24000 x^{3} - 6000 x^{2} + 500 x\right ) \log {\left (x + 1 \right )}^{3} + \left (160000 x^{4} - 160000 x^{3} + 60000 x^{2} - 10000 x + 625\right ) \log {\left (x + 1 \right )}^{4}} \] Input:
integrate((-320000000*x**5+396000000*x**4-201000000*x**3+52250000*x**2-693 7500*x+375000)/((3200000*x**6-800000*x**5-2000000*x**4+1500000*x**3-437500 *x**2+59375*x-3125)*ln(1+x)**5+(-800000*x**6+500000*x**4-250000*x**3+46875 *x**2-3125*x)*ln(1+x)**4+(80000*x**6+20000*x**5-45000*x**4+13750*x**3-1250 *x**2)*ln(1+x)**3+(-4000*x**6-2000*x**5+1750*x**4-250*x**3)*ln(1+x)**2+(10 0*x**6+75*x**5-25*x**4)*ln(1+x)-x**6-x**5),x)
Output:
(4000000*x**4 - 4000000*x**3 + 1500000*x**2 - 250000*x + 15625)/(x**4 + (- 80*x**4 + 20*x**3)*log(x + 1) + (2400*x**4 - 1200*x**3 + 150*x**2)*log(x + 1)**2 + (-32000*x**4 + 24000*x**3 - 6000*x**2 + 500*x)*log(x + 1)**3 + (1 60000*x**4 - 160000*x**3 + 60000*x**2 - 10000*x + 625)*log(x + 1)**4)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.52 \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\frac {15625 \, {\left (256 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 16 \, x + 1\right )}}{625 \, {\left (256 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 16 \, x + 1\right )} \log \left (x + 1\right )^{4} + x^{4} - 500 \, {\left (64 \, x^{4} - 48 \, x^{3} + 12 \, x^{2} - x\right )} \log \left (x + 1\right )^{3} + 150 \, {\left (16 \, x^{4} - 8 \, x^{3} + x^{2}\right )} \log \left (x + 1\right )^{2} - 20 \, {\left (4 \, x^{4} - x^{3}\right )} \log \left (x + 1\right )} \] Input:
integrate((-320000000*x^5+396000000*x^4-201000000*x^3+52250000*x^2-6937500 *x+375000)/((3200000*x^6-800000*x^5-2000000*x^4+1500000*x^3-437500*x^2+593 75*x-3125)*log(1+x)^5+(-800000*x^6+500000*x^4-250000*x^3+46875*x^2-3125*x) *log(1+x)^4+(80000*x^6+20000*x^5-45000*x^4+13750*x^3-1250*x^2)*log(1+x)^3+ (-4000*x^6-2000*x^5+1750*x^4-250*x^3)*log(1+x)^2+(100*x^6+75*x^5-25*x^4)*l og(1+x)-x^6-x^5),x, algorithm="maxima")
Output:
15625*(256*x^4 - 256*x^3 + 96*x^2 - 16*x + 1)/(625*(256*x^4 - 256*x^3 + 96 *x^2 - 16*x + 1)*log(x + 1)^4 + x^4 - 500*(64*x^4 - 48*x^3 + 12*x^2 - x)*l og(x + 1)^3 + 150*(16*x^4 - 8*x^3 + x^2)*log(x + 1)^2 - 20*(4*x^4 - x^3)*l og(x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 10.26 \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\frac {15625 \, {\left (20480 \, x^{6} - 30464 \, x^{5} + 19200 \, x^{4} - 6560 \, x^{3} + 1280 \, x^{2} - 135 \, x + 6\right )}}{12800000 \, x^{6} \log \left (x + 1\right )^{4} - 2560000 \, x^{6} \log \left (x + 1\right )^{3} - 19040000 \, x^{5} \log \left (x + 1\right )^{4} + 192000 \, x^{6} \log \left (x + 1\right )^{2} + 3168000 \, x^{5} \log \left (x + 1\right )^{3} + 12000000 \, x^{4} \log \left (x + 1\right )^{4} - 6400 \, x^{6} \log \left (x + 1\right ) - 189600 \, x^{5} \log \left (x + 1\right )^{2} - 1608000 \, x^{4} \log \left (x + 1\right )^{3} - 4100000 \, x^{3} \log \left (x + 1\right )^{4} + 80 \, x^{6} + 4720 \, x^{5} \log \left (x + 1\right ) + 73200 \, x^{4} \log \left (x + 1\right )^{2} + 418000 \, x^{3} \log \left (x + 1\right )^{3} + 800000 \, x^{2} \log \left (x + 1\right )^{4} - 39 \, x^{5} - 1260 \, x^{4} \log \left (x + 1\right ) - 13050 \, x^{3} \log \left (x + 1\right )^{2} - 55500 \, x^{2} \log \left (x + 1\right )^{3} - 84375 \, x \log \left (x + 1\right )^{4} + 6 \, x^{4} + 120 \, x^{3} \log \left (x + 1\right ) + 900 \, x^{2} \log \left (x + 1\right )^{2} + 3000 \, x \log \left (x + 1\right )^{3} + 3750 \, \log \left (x + 1\right )^{4}} \] Input:
integrate((-320000000*x^5+396000000*x^4-201000000*x^3+52250000*x^2-6937500 *x+375000)/((3200000*x^6-800000*x^5-2000000*x^4+1500000*x^3-437500*x^2+593 75*x-3125)*log(1+x)^5+(-800000*x^6+500000*x^4-250000*x^3+46875*x^2-3125*x) *log(1+x)^4+(80000*x^6+20000*x^5-45000*x^4+13750*x^3-1250*x^2)*log(1+x)^3+ (-4000*x^6-2000*x^5+1750*x^4-250*x^3)*log(1+x)^2+(100*x^6+75*x^5-25*x^4)*l og(1+x)-x^6-x^5),x, algorithm="giac")
Output:
15625*(20480*x^6 - 30464*x^5 + 19200*x^4 - 6560*x^3 + 1280*x^2 - 135*x + 6 )/(12800000*x^6*log(x + 1)^4 - 2560000*x^6*log(x + 1)^3 - 19040000*x^5*log (x + 1)^4 + 192000*x^6*log(x + 1)^2 + 3168000*x^5*log(x + 1)^3 + 12000000* x^4*log(x + 1)^4 - 6400*x^6*log(x + 1) - 189600*x^5*log(x + 1)^2 - 1608000 *x^4*log(x + 1)^3 - 4100000*x^3*log(x + 1)^4 + 80*x^6 + 4720*x^5*log(x + 1 ) + 73200*x^4*log(x + 1)^2 + 418000*x^3*log(x + 1)^3 + 800000*x^2*log(x + 1)^4 - 39*x^5 - 1260*x^4*log(x + 1) - 13050*x^3*log(x + 1)^2 - 55500*x^2*l og(x + 1)^3 - 84375*x*log(x + 1)^4 + 6*x^4 + 120*x^3*log(x + 1) + 900*x^2* log(x + 1)^2 + 3000*x*log(x + 1)^3 + 3750*log(x + 1)^4)
Timed out. \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\int \frac {320000000\,x^5-396000000\,x^4+201000000\,x^3-52250000\,x^2+6937500\,x-375000}{{\ln \left (x+1\right )}^4\,\left (800000\,x^6-500000\,x^4+250000\,x^3-46875\,x^2+3125\,x\right )+{\ln \left (x+1\right )}^2\,\left (4000\,x^6+2000\,x^5-1750\,x^4+250\,x^3\right )+{\ln \left (x+1\right )}^5\,\left (-3200000\,x^6+800000\,x^5+2000000\,x^4-1500000\,x^3+437500\,x^2-59375\,x+3125\right )-{\ln \left (x+1\right )}^3\,\left (80000\,x^6+20000\,x^5-45000\,x^4+13750\,x^3-1250\,x^2\right )-\ln \left (x+1\right )\,\left (100\,x^6+75\,x^5-25\,x^4\right )+x^5+x^6} \,d x \] Input:
int((6937500*x - 52250000*x^2 + 201000000*x^3 - 396000000*x^4 + 320000000* x^5 - 375000)/(log(x + 1)^4*(3125*x - 46875*x^2 + 250000*x^3 - 500000*x^4 + 800000*x^6) + log(x + 1)^2*(250*x^3 - 1750*x^4 + 2000*x^5 + 4000*x^6) + log(x + 1)^5*(437500*x^2 - 59375*x - 1500000*x^3 + 2000000*x^4 + 800000*x^ 5 - 3200000*x^6 + 3125) - log(x + 1)^3*(13750*x^3 - 1250*x^2 - 45000*x^4 + 20000*x^5 + 80000*x^6) - log(x + 1)*(75*x^5 - 25*x^4 + 100*x^6) + x^5 + x ^6),x)
Output:
int((6937500*x - 52250000*x^2 + 201000000*x^3 - 396000000*x^4 + 320000000* x^5 - 375000)/(log(x + 1)^4*(3125*x - 46875*x^2 + 250000*x^3 - 500000*x^4 + 800000*x^6) + log(x + 1)^2*(250*x^3 - 1750*x^4 + 2000*x^5 + 4000*x^6) + log(x + 1)^5*(437500*x^2 - 59375*x - 1500000*x^3 + 2000000*x^4 + 800000*x^ 5 - 3200000*x^6 + 3125) - log(x + 1)^3*(13750*x^3 - 1250*x^2 - 45000*x^4 + 20000*x^5 + 80000*x^6) - log(x + 1)*(75*x^5 - 25*x^4 + 100*x^6) + x^5 + x ^6), x)
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 6.33 \[ \int \frac {375000-6937500 x+52250000 x^2-201000000 x^3+396000000 x^4-320000000 x^5}{-x^5-x^6+\left (-25 x^4+75 x^5+100 x^6\right ) \log (1+x)+\left (-250 x^3+1750 x^4-2000 x^5-4000 x^6\right ) \log ^2(1+x)+\left (-1250 x^2+13750 x^3-45000 x^4+20000 x^5+80000 x^6\right ) \log ^3(1+x)+\left (-3125 x+46875 x^2-250000 x^3+500000 x^4-800000 x^6\right ) \log ^4(1+x)+\left (-3125+59375 x-437500 x^2+1500000 x^3-2000000 x^4-800000 x^5+3200000 x^6\right ) \log ^5(1+x)} \, dx=\frac {4000000 x^{4}-4000000 x^{3}+1500000 x^{2}-250000 x +15625}{160000 \mathrm {log}\left (x +1\right )^{4} x^{4}-160000 \mathrm {log}\left (x +1\right )^{4} x^{3}+60000 \mathrm {log}\left (x +1\right )^{4} x^{2}-10000 \mathrm {log}\left (x +1\right )^{4} x +625 \mathrm {log}\left (x +1\right )^{4}-32000 \mathrm {log}\left (x +1\right )^{3} x^{4}+24000 \mathrm {log}\left (x +1\right )^{3} x^{3}-6000 \mathrm {log}\left (x +1\right )^{3} x^{2}+500 \mathrm {log}\left (x +1\right )^{3} x +2400 \mathrm {log}\left (x +1\right )^{2} x^{4}-1200 \mathrm {log}\left (x +1\right )^{2} x^{3}+150 \mathrm {log}\left (x +1\right )^{2} x^{2}-80 \,\mathrm {log}\left (x +1\right ) x^{4}+20 \,\mathrm {log}\left (x +1\right ) x^{3}+x^{4}} \] Input:
int((-320000000*x^5+396000000*x^4-201000000*x^3+52250000*x^2-6937500*x+375 000)/((3200000*x^6-800000*x^5-2000000*x^4+1500000*x^3-437500*x^2+59375*x-3 125)*log(1+x)^5+(-800000*x^6+500000*x^4-250000*x^3+46875*x^2-3125*x)*log(1 +x)^4+(80000*x^6+20000*x^5-45000*x^4+13750*x^3-1250*x^2)*log(1+x)^3+(-4000 *x^6-2000*x^5+1750*x^4-250*x^3)*log(1+x)^2+(100*x^6+75*x^5-25*x^4)*log(1+x )-x^6-x^5),x)
Output:
(15625*(256*x**4 - 256*x**3 + 96*x**2 - 16*x + 1))/(160000*log(x + 1)**4*x **4 - 160000*log(x + 1)**4*x**3 + 60000*log(x + 1)**4*x**2 - 10000*log(x + 1)**4*x + 625*log(x + 1)**4 - 32000*log(x + 1)**3*x**4 + 24000*log(x + 1) **3*x**3 - 6000*log(x + 1)**3*x**2 + 500*log(x + 1)**3*x + 2400*log(x + 1) **2*x**4 - 1200*log(x + 1)**2*x**3 + 150*log(x + 1)**2*x**2 - 80*log(x + 1 )*x**4 + 20*log(x + 1)*x**3 + x**4)