Integrand size = 125, antiderivative size = 28 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{16} x^2 (10-2 x-\log (4))^2 \left (x+4 \log ^2(5)\right )^2 \] Output:
25/16*(4*ln(5)^2+x)^2*(10-2*x-2*ln(2))^2*x^2
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{16} x^2 (-10+2 x+\log (4))^2 \left (x+4 \log ^2(5)\right )^2 \] Input:
Integrate[(2500*x^3 - 1250*x^4 + 150*x^5 + (-500*x^3 + 125*x^4)*Log[4] + 2 5*x^3*Log[4]^2 + (15000*x^2 - 8000*x^3 + 1000*x^4 + (-3000*x^2 + 800*x^3)* Log[4] + 150*x^2*Log[4]^2)*Log[5]^2 + (20000*x - 12000*x^2 + 1600*x^3 + (- 4000*x + 1200*x^2)*Log[4] + 200*x*Log[4]^2)*Log[5]^4)/4,x]
Output:
(25*x^2*(-10 + 2*x + Log[4])^2*(x + 4*Log[5]^2)^2)/16
Leaf count is larger than twice the leaf count of optimal. \(152\) vs. \(2(28)=56\).
Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 5.43, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 27, 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.
| \(\displaystyle \int \frac {1}{4} \left (150 x^5-1250 x^4+2500 x^3+25 x^3 \log ^2(4)+\left (125 x^4-500 x^3\right ) \log (4)+\log ^4(5) \left (1600 x^3-12000 x^2+\left (1200 x^2-4000 x\right ) \log (4)+20000 x+200 x \log ^2(4)\right )+\log ^2(5) \left (1000 x^4-8000 x^3+15000 x^2+150 x^2 \log ^2(4)+\left (800 x^3-3000 x^2\right ) \log (4)\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {1}{4} \left (150 x^5-1250 x^4+x^3 \left (2500+25 \log ^2(4)\right )+\left (125 x^4-500 x^3\right ) \log (4)+\log ^4(5) \left (1600 x^3-12000 x^2+\left (1200 x^2-4000 x\right ) \log (4)+20000 x+200 x \log ^2(4)\right )+\log ^2(5) \left (1000 x^4-8000 x^3+15000 x^2+150 x^2 \log ^2(4)+\left (800 x^3-3000 x^2\right ) \log (4)\right )\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \left (150 x^5-1250 x^4+25 \left (100+\log ^2(4)\right ) x^3+200 \left (8 x^3-60 x^2+\log ^2(4) x+100 x-2 \left (10 x-3 x^2\right ) \log (4)\right ) \log ^4(5)+50 \left (20 x^4-160 x^3+3 \log ^2(4) x^2+300 x^2-4 \left (15 x^2-4 x^3\right ) \log (4)\right ) \log ^2(5)-125 \left (4 x^3-x^4\right ) \log (4)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (25 x^6-250 x^5+200 x^5 \log ^2(5)+25 x^5 \log (4)+400 x^4 \log ^4(5)+200 x^4 \log (4) \log ^2(5)-2000 x^4 \log ^2(5)+\frac {25}{4} x^4 \left (100+\log ^2(4)\right )-125 x^4 \log (4)+400 x^3 \log (4) \log ^4(5)-4000 x^3 \log ^4(5)+50 x^3 \left (100+\log ^2(4)\right ) \log ^2(5)-1000 x^3 \log (4) \log ^2(5)-2000 x^2 \log (4) \log ^4(5)+100 x^2 \left (100+\log ^2(4)\right ) \log ^4(5)\right )\) |
Input:
Int[(2500*x^3 - 1250*x^4 + 150*x^5 + (-500*x^3 + 125*x^4)*Log[4] + 25*x^3* Log[4]^2 + (15000*x^2 - 8000*x^3 + 1000*x^4 + (-3000*x^2 + 800*x^3)*Log[4] + 150*x^2*Log[4]^2)*Log[5]^2 + (20000*x - 12000*x^2 + 1600*x^3 + (-4000*x + 1200*x^2)*Log[4] + 200*x*Log[4]^2)*Log[5]^4)/4,x]
Output:
(-250*x^5 + 25*x^6 - 125*x^4*Log[4] + 25*x^5*Log[4] + (25*x^4*(100 + Log[4 ]^2))/4 - 2000*x^4*Log[5]^2 + 200*x^5*Log[5]^2 - 1000*x^3*Log[4]*Log[5]^2 + 200*x^4*Log[4]*Log[5]^2 + 50*x^3*(100 + Log[4]^2)*Log[5]^2 - 4000*x^3*Lo g[5]^4 + 400*x^4*Log[5]^4 - 2000*x^2*Log[4]*Log[5]^4 + 400*x^3*Log[4]*Log[ 5]^4 + 100*x^2*(100 + Log[4]^2)*Log[5]^4)/4
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(92\) vs. \(2(26)=52\).
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.32
| method | result | size |
| gosper | \(\frac {25 \left (4 \ln \left (5\right )^{2}+x \right ) \left (4 \ln \left (5\right )^{2} \ln \left (2\right )^{2}+8 \ln \left (5\right )^{2} \ln \left (2\right ) x +4 x^{2} \ln \left (5\right )^{2}-40 \ln \left (2\right ) \ln \left (5\right )^{2}-40 x \ln \left (5\right )^{2}+x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+x^{3}+100 \ln \left (5\right )^{2}-10 x \ln \left (2\right )-10 x^{2}+25 x \right ) x^{2}}{4}\) | \(93\) |
| norman | \(\left (50 \ln \left (5\right )^{2}+\frac {25 \ln \left (2\right )}{2}-\frac {125}{2}\right ) x^{5}+\left (100 \ln \left (5\right )^{4} \ln \left (2\right )^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right )+2500 \ln \left (5\right )^{4}\right ) x^{2}+\left (200 \ln \left (5\right )^{4} \ln \left (2\right )-1000 \ln \left (5\right )^{4}+50 \ln \left (5\right )^{2} \ln \left (2\right )^{2}-500 \ln \left (2\right ) \ln \left (5\right )^{2}+1250 \ln \left (5\right )^{2}\right ) x^{3}+\left (100 \ln \left (5\right )^{4}+100 \ln \left (2\right ) \ln \left (5\right )^{2}-500 \ln \left (5\right )^{2}+\frac {25 \ln \left (2\right )^{2}}{4}-\frac {125 \ln \left (2\right )}{2}+\frac {625}{4}\right ) x^{4}+\frac {25 x^{6}}{4}\) | \(131\) |
| default | \(\frac {25 x^{6}}{4}+\frac {5 \left (20 \ln \left (5\right )^{2}+5 \ln \left (2\right )-25\right ) x^{5}}{2}+\frac {25 \left (12 \ln \left (5\right )^{2} \left (-5+\ln \left (2\right )\right )+\left (4 \ln \left (5\right )^{2}+\ln \left (2\right )-5\right ) \left (8 \ln \left (5\right )^{2}+2 \ln \left (2\right )-10\right )+4 \ln \left (2\right ) \ln \left (5\right )^{2}-20 \ln \left (5\right )^{2}\right ) x^{4}}{8}+\frac {25 \left (4 \ln \left (5\right )^{2} \left (-5+\ln \left (2\right )\right ) \left (8 \ln \left (5\right )^{2}+2 \ln \left (2\right )-10\right )+\left (4 \ln \left (5\right )^{2}+\ln \left (2\right )-5\right ) \left (4 \ln \left (2\right ) \ln \left (5\right )^{2}-20 \ln \left (5\right )^{2}\right )\right ) x^{3}}{6}+25 \ln \left (5\right )^{2} \left (-5+\ln \left (2\right )\right ) \left (4 \ln \left (2\right ) \ln \left (5\right )^{2}-20 \ln \left (5\right )^{2}\right ) x^{2}\) | \(159\) |
| risch | \(100 \ln \left (5\right )^{4} x^{4}+200 \ln \left (5\right )^{4} \ln \left (2\right ) x^{3}-1000 \ln \left (5\right )^{4} x^{3}+100 \ln \left (5\right )^{4} \ln \left (2\right )^{2} x^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right ) x^{2}+2500 \ln \left (5\right )^{4} x^{2}+50 \ln \left (5\right )^{2} x^{5}+100 \ln \left (5\right )^{2} \ln \left (2\right ) x^{4}-500 \ln \left (5\right )^{2} x^{4}+50 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{3}-500 \ln \left (2\right ) \ln \left (5\right )^{2} x^{3}+1250 x^{3} \ln \left (5\right )^{2}+\frac {25 x^{4} \ln \left (2\right )^{2}}{4}+\frac {25 x^{5} \ln \left (2\right )}{2}-\frac {125 x^{4} \ln \left (2\right )}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\) | \(164\) |
| parallelrisch | \(100 \ln \left (5\right )^{4} x^{4}+200 \ln \left (5\right )^{4} \ln \left (2\right ) x^{3}-1000 \ln \left (5\right )^{4} x^{3}+100 \ln \left (5\right )^{4} \ln \left (2\right )^{2} x^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right ) x^{2}+2500 \ln \left (5\right )^{4} x^{2}+50 \ln \left (5\right )^{2} x^{5}+100 \ln \left (5\right )^{2} \ln \left (2\right ) x^{4}-500 \ln \left (5\right )^{2} x^{4}+50 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{3}-500 \ln \left (2\right ) \ln \left (5\right )^{2} x^{3}+1250 x^{3} \ln \left (5\right )^{2}+\frac {25 x^{4} \ln \left (2\right )^{2}}{4}+\frac {25 x^{5} \ln \left (2\right )}{2}-\frac {125 x^{4} \ln \left (2\right )}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\) | \(164\) |
| parts | \(100 \ln \left (5\right )^{4} x^{4}+200 \ln \left (5\right )^{4} \ln \left (2\right ) x^{3}-1000 \ln \left (5\right )^{4} x^{3}+100 \ln \left (5\right )^{4} \ln \left (2\right )^{2} x^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right ) x^{2}+2500 \ln \left (5\right )^{4} x^{2}+50 \ln \left (5\right )^{2} x^{5}+100 \ln \left (5\right )^{2} \ln \left (2\right ) x^{4}-500 \ln \left (5\right )^{2} x^{4}+50 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{3}-500 \ln \left (2\right ) \ln \left (5\right )^{2} x^{3}+1250 x^{3} \ln \left (5\right )^{2}+\frac {25 x^{4} \ln \left (2\right )^{2}}{4}+\frac {25 x^{5} \ln \left (2\right )}{2}-\frac {125 x^{4} \ln \left (2\right )}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\) | \(164\) |
Input:
int(1/4*(800*x*ln(2)^2+2*(1200*x^2-4000*x)*ln(2)+1600*x^3-12000*x^2+20000* x)*ln(5)^4+1/4*(600*x^2*ln(2)^2+2*(800*x^3-3000*x^2)*ln(2)+1000*x^4-8000*x ^3+15000*x^2)*ln(5)^2+25*x^3*ln(2)^2+1/2*(125*x^4-500*x^3)*ln(2)+75/2*x^5- 625/2*x^4+625*x^3,x,method=_RETURNVERBOSE)
Output:
25/4*(4*ln(5)^2+x)*(4*ln(5)^2*ln(2)^2+8*ln(5)^2*ln(2)*x+4*x^2*ln(5)^2-40*l n(2)*ln(5)^2-40*x*ln(5)^2+x*ln(2)^2+2*x^2*ln(2)+x^3+100*ln(5)^2-10*x*ln(2) -10*x^2+25*x)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \left (2\right )^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \left (2\right )^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (5\right )^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \left (2\right )^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (5\right )^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \left (2\right ) \] Input:
integrate(1/4*(800*x*log(2)^2+2*(1200*x^2-4000*x)*log(2)+1600*x^3-12000*x^ 2+20000*x)*log(5)^4+1/4*(600*x^2*log(2)^2+2*(800*x^3-3000*x^2)*log(2)+1000 *x^4-8000*x^3+15000*x^2)*log(5)^2+25*x^3*log(2)^2+1/2*(125*x^4-500*x^3)*lo g(2)+75/2*x^5-625/2*x^4+625*x^3,x, algorithm="fricas")
Output:
25/4*x^6 + 25/4*x^4*log(2)^2 - 125/2*x^5 + 100*(x^4 + x^2*log(2)^2 - 10*x^ 3 + 25*x^2 + 2*(x^3 - 5*x^2)*log(2))*log(5)^4 + 625/4*x^4 + 50*(x^5 + x^3* log(2)^2 - 10*x^4 + 25*x^3 + 2*(x^4 - 5*x^3)*log(2))*log(5)^2 + 25/2*(x^5 - 5*x^4)*log(2)
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (27) = 54\).
Time = 0.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 5.46 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25 x^{6}}{4} + x^{5} \left (- \frac {125}{2} + \frac {25 \log {\left (2 \right )}}{2} + 50 \log {\left (5 \right )}^{2}\right ) + x^{4} \left (- 500 \log {\left (5 \right )}^{2} - \frac {125 \log {\left (2 \right )}}{2} + \frac {25 \log {\left (2 \right )}^{2}}{4} + \frac {625}{4} + 100 \log {\left (2 \right )} \log {\left (5 \right )}^{2} + 100 \log {\left (5 \right )}^{4}\right ) + x^{3} \left (- 1000 \log {\left (5 \right )}^{4} - 500 \log {\left (2 \right )} \log {\left (5 \right )}^{2} + 50 \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{2} + 200 \log {\left (2 \right )} \log {\left (5 \right )}^{4} + 1250 \log {\left (5 \right )}^{2}\right ) + x^{2} \left (- 1000 \log {\left (2 \right )} \log {\left (5 \right )}^{4} + 100 \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{4} + 2500 \log {\left (5 \right )}^{4}\right ) \] Input:
integrate(1/4*(800*x*ln(2)**2+2*(1200*x**2-4000*x)*ln(2)+1600*x**3-12000*x **2+20000*x)*ln(5)**4+1/4*(600*x**2*ln(2)**2+2*(800*x**3-3000*x**2)*ln(2)+ 1000*x**4-8000*x**3+15000*x**2)*ln(5)**2+25*x**3*ln(2)**2+1/2*(125*x**4-50 0*x**3)*ln(2)+75/2*x**5-625/2*x**4+625*x**3,x)
Output:
25*x**6/4 + x**5*(-125/2 + 25*log(2)/2 + 50*log(5)**2) + x**4*(-500*log(5) **2 - 125*log(2)/2 + 25*log(2)**2/4 + 625/4 + 100*log(2)*log(5)**2 + 100*l og(5)**4) + x**3*(-1000*log(5)**4 - 500*log(2)*log(5)**2 + 50*log(2)**2*lo g(5)**2 + 200*log(2)*log(5)**4 + 1250*log(5)**2) + x**2*(-1000*log(2)*log( 5)**4 + 100*log(2)**2*log(5)**4 + 2500*log(5)**4)
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (22) = 44\).
Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \left (2\right )^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \left (2\right )^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (5\right )^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \left (2\right )^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (5\right )^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \left (2\right ) \] Input:
integrate(1/4*(800*x*log(2)^2+2*(1200*x^2-4000*x)*log(2)+1600*x^3-12000*x^ 2+20000*x)*log(5)^4+1/4*(600*x^2*log(2)^2+2*(800*x^3-3000*x^2)*log(2)+1000 *x^4-8000*x^3+15000*x^2)*log(5)^2+25*x^3*log(2)^2+1/2*(125*x^4-500*x^3)*lo g(2)+75/2*x^5-625/2*x^4+625*x^3,x, algorithm="maxima")
Output:
25/4*x^6 + 25/4*x^4*log(2)^2 - 125/2*x^5 + 100*(x^4 + x^2*log(2)^2 - 10*x^ 3 + 25*x^2 + 2*(x^3 - 5*x^2)*log(2))*log(5)^4 + 625/4*x^4 + 50*(x^5 + x^3* log(2)^2 - 10*x^4 + 25*x^3 + 2*(x^4 - 5*x^3)*log(2))*log(5)^2 + 25/2*(x^5 - 5*x^4)*log(2)
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \left (2\right )^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \left (2\right )^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (5\right )^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \left (2\right )^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (5\right )^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \left (2\right ) \] Input:
integrate(1/4*(800*x*log(2)^2+2*(1200*x^2-4000*x)*log(2)+1600*x^3-12000*x^ 2+20000*x)*log(5)^4+1/4*(600*x^2*log(2)^2+2*(800*x^3-3000*x^2)*log(2)+1000 *x^4-8000*x^3+15000*x^2)*log(5)^2+25*x^3*log(2)^2+1/2*(125*x^4-500*x^3)*lo g(2)+75/2*x^5-625/2*x^4+625*x^3,x, algorithm="giac")
Output:
25/4*x^6 + 25/4*x^4*log(2)^2 - 125/2*x^5 + 100*(x^4 + x^2*log(2)^2 - 10*x^ 3 + 25*x^2 + 2*(x^3 - 5*x^2)*log(2))*log(5)^4 + 625/4*x^4 + 50*(x^5 + x^3* log(2)^2 - 10*x^4 + 25*x^3 + 2*(x^4 - 5*x^3)*log(2))*log(5)^2 + 25/2*(x^5 - 5*x^4)*log(2)
Time = 2.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.43 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25\,x^6}{4}+\left (\frac {25\,\ln \left (2\right )}{2}+50\,{\ln \left (5\right )}^2-\frac {125}{2}\right )\,x^5+\left (100\,\ln \left (2\right )\,{\ln \left (5\right )}^2-\frac {125\,\ln \left (2\right )}{2}+\frac {25\,{\ln \left (2\right )}^2}{4}-500\,{\ln \left (5\right )}^2+100\,{\ln \left (5\right )}^4+\frac {625}{4}\right )\,x^4+50\,{\ln \left (5\right )}^2\,\left (\ln \left (2\right )-5\right )\,\left (\ln \left (2\right )+4\,{\ln \left (5\right )}^2-5\right )\,x^3+100\,{\ln \left (5\right )}^4\,{\left (\ln \left (2\right )-5\right )}^2\,x^2 \] Input:
int(25*x^3*log(2)^2 - (log(2)*(500*x^3 - 125*x^4))/2 + (log(5)^4*(20000*x - 2*log(2)*(4000*x - 1200*x^2) + 800*x*log(2)^2 - 12000*x^2 + 1600*x^3))/4 + (log(5)^2*(600*x^2*log(2)^2 - 2*log(2)*(3000*x^2 - 800*x^3) + 15000*x^2 - 8000*x^3 + 1000*x^4))/4 + 625*x^3 - (625*x^4)/2 + (75*x^5)/2,x)
Output:
x^4*(100*log(2)*log(5)^2 - (125*log(2))/2 + (25*log(2)^2)/4 - 500*log(5)^2 + 100*log(5)^4 + 625/4) + x^5*((25*log(2))/2 + 50*log(5)^2 - 125/2) + (25 *x^6)/4 + 100*x^2*log(5)^4*(log(2) - 5)^2 + 50*x^3*log(5)^2*(log(2) - 5)*( log(2) + 4*log(5)^2 - 5)
Time = 0.17 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25 x^{2} \left (16 \mathrm {log}\left (5\right )^{4} \mathrm {log}\left (2\right )^{2}+32 \mathrm {log}\left (5\right )^{4} \mathrm {log}\left (2\right ) x -160 \mathrm {log}\left (5\right )^{4} \mathrm {log}\left (2\right )+16 \mathrm {log}\left (5\right )^{4} x^{2}-160 \mathrm {log}\left (5\right )^{4} x +400 \mathrm {log}\left (5\right )^{4}+8 \mathrm {log}\left (5\right )^{2} \mathrm {log}\left (2\right )^{2} x +16 \mathrm {log}\left (5\right )^{2} \mathrm {log}\left (2\right ) x^{2}-80 \mathrm {log}\left (5\right )^{2} \mathrm {log}\left (2\right ) x +8 \mathrm {log}\left (5\right )^{2} x^{3}-80 \mathrm {log}\left (5\right )^{2} x^{2}+200 \mathrm {log}\left (5\right )^{2} x +\mathrm {log}\left (2\right )^{2} x^{2}+2 \,\mathrm {log}\left (2\right ) x^{3}-10 \,\mathrm {log}\left (2\right ) x^{2}+x^{4}-10 x^{3}+25 x^{2}\right )}{4} \] Input:
int(1/4*(800*x*log(2)^2+2*(1200*x^2-4000*x)*log(2)+1600*x^3-12000*x^2+2000 0*x)*log(5)^4+1/4*(600*x^2*log(2)^2+2*(800*x^3-3000*x^2)*log(2)+1000*x^4-8 000*x^3+15000*x^2)*log(5)^2+25*x^3*log(2)^2+1/2*(125*x^4-500*x^3)*log(2)+7 5/2*x^5-625/2*x^4+625*x^3,x)
Output:
(25*x**2*(16*log(5)**4*log(2)**2 + 32*log(5)**4*log(2)*x - 160*log(5)**4*l og(2) + 16*log(5)**4*x**2 - 160*log(5)**4*x + 400*log(5)**4 + 8*log(5)**2* log(2)**2*x + 16*log(5)**2*log(2)*x**2 - 80*log(5)**2*log(2)*x + 8*log(5)* *2*x**3 - 80*log(5)**2*x**2 + 200*log(5)**2*x + log(2)**2*x**2 + 2*log(2)* x**3 - 10*log(2)*x**2 + x**4 - 10*x**3 + 25*x**2))/4