Integrand size = 72, antiderivative size = 27 \[ \int \frac {1}{16} \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx=16+x \left (9+\left (\frac {x}{4}+2 (3-x) (7+\log (5))^2\right )^2\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(27)=54\).
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {1}{16} \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx=-3 x^2 (7+\log (5))^2 \left (391+112 \log (5)+8 \log ^2(5)\right )+\frac {1}{16} x^3 \left (391+112 \log (5)+8 \log ^2(5)\right )^2+9 x \left (9605+5488 \log (5)+1176 \log ^2(5)+112 \log ^3(5)+4 \log ^4(5)\right ) \]
Integrate[(1383120 - 1839264*x + 458643*x^2 + (790272 - 1052352*x + 262752 *x^2)*Log[5] + (169344 - 225696*x + 56400*x^2)*Log[5]^2 + (16128 - 21504*x + 5376*x^2)*Log[5]^3 + (576 - 768*x + 192*x^2)*Log[5]^4)/16,x]
-3*x^2*(7 + Log[5])^2*(391 + 112*Log[5] + 8*Log[5]^2) + (x^3*(391 + 112*Lo g[5] + 8*Log[5]^2)^2)/16 + 9*x*(9605 + 5488*Log[5] + 1176*Log[5]^2 + 112*L og[5]^3 + 4*Log[5]^4)
Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(27)=54\).
Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.67, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {27, 6, 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.
\(\displaystyle \int \frac {1}{16} \left (458643 x^2+\left (192 x^2-768 x+576\right ) \log ^4(5)+\left (5376 x^2-21504 x+16128\right ) \log ^3(5)+\left (56400 x^2-225696 x+169344\right ) \log ^2(5)+\left (262752 x^2-1052352 x+790272\right ) \log (5)-1839264 x+1383120\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \int \left (458643 x^2-1839264 x+192 \left (x^2-4 x+3\right ) \log ^4(5)+5376 \left (x^2-4 x+3\right ) \log ^3(5)+48 \left (1175 x^2-4702 x+3528\right ) \log ^2(5)+672 \left (391 x^2-1566 x+1176\right ) \log (5)+1383120\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \frac {1}{16} \int \left (458643 x^2-1839264 x+\left (x^2-4 x+3\right ) \left (5376 \log ^3(5)+192 \log ^4(5)\right )+48 \left (1175 x^2-4702 x+3528\right ) \log ^2(5)+672 \left (391 x^2-1566 x+1176\right ) \log (5)+1383120\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{16} \left (152881 x^3+64 x^3 \log ^3(5) (28+\log (5))+18800 x^3 \log ^2(5)+87584 x^3 \log (5)-919632 x^2-384 x^2 \log ^3(5) (28+\log (5))-112848 x^2 \log ^2(5)-526176 x^2 \log (5)+1383120 x+576 x \log ^3(5) (28+\log (5))+169344 x \log ^2(5)+790272 x \log (5)\right )\) |
Int[(1383120 - 1839264*x + 458643*x^2 + (790272 - 1052352*x + 262752*x^2)* Log[5] + (169344 - 225696*x + 56400*x^2)*Log[5]^2 + (16128 - 21504*x + 537 6*x^2)*Log[5]^3 + (576 - 768*x + 192*x^2)*Log[5]^4)/16,x]
(1383120*x - 919632*x^2 + 152881*x^3 + 790272*x*Log[5] - 526176*x^2*Log[5] + 87584*x^3*Log[5] + 169344*x*Log[5]^2 - 112848*x^2*Log[5]^2 + 18800*x^3* Log[5]^2 + 576*x*Log[5]^3*(28 + Log[5]) - 384*x^2*Log[5]^3*(28 + Log[5]) + 64*x^3*Log[5]^3*(28 + Log[5]))/16
3.5.4.3.1 Defintions of rubi rules used
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. \(83\) vs. \(2(24)=48\).
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.11
method | result | size |
norman | \(\left (-24 \ln \left (5\right )^{4}-672 \ln \left (5\right )^{3}-7053 \ln \left (5\right )^{2}-32886 \ln \left (5\right )-57477\right ) x^{2}+\left (4 \ln \left (5\right )^{4}+112 \ln \left (5\right )^{3}+1175 \ln \left (5\right )^{2}+5474 \ln \left (5\right )+\frac {152881}{16}\right ) x^{3}+\left (36 \ln \left (5\right )^{4}+1008 \ln \left (5\right )^{3}+10584 \ln \left (5\right )^{2}+49392 \ln \left (5\right )+86445\right ) x\) | \(84\) |
gosper | \(\frac {x \left (64 \ln \left (5\right )^{4} x^{2}-384 x \ln \left (5\right )^{4}+1792 x^{2} \ln \left (5\right )^{3}+576 \ln \left (5\right )^{4}-10752 \ln \left (5\right )^{3} x +18800 x^{2} \ln \left (5\right )^{2}+16128 \ln \left (5\right )^{3}-112848 x \ln \left (5\right )^{2}+87584 x^{2} \ln \left (5\right )+169344 \ln \left (5\right )^{2}-526176 x \ln \left (5\right )+152881 x^{2}+790272 \ln \left (5\right )-919632 x +1383120\right )}{16}\) | \(96\) |
default | \(4 x^{3} \ln \left (5\right )^{4}-24 \ln \left (5\right )^{4} x^{2}+112 x^{3} \ln \left (5\right )^{3}+36 x \ln \left (5\right )^{4}-672 x^{2} \ln \left (5\right )^{3}+1175 x^{3} \ln \left (5\right )^{2}+1008 \ln \left (5\right )^{3} x -7053 x^{2} \ln \left (5\right )^{2}+5474 x^{3} \ln \left (5\right )+10584 x \ln \left (5\right )^{2}-32886 x^{2} \ln \left (5\right )+\frac {152881 x^{3}}{16}+49392 x \ln \left (5\right )-57477 x^{2}+86445 x\) | \(109\) |
risch | \(4 x^{3} \ln \left (5\right )^{4}-24 \ln \left (5\right )^{4} x^{2}+112 x^{3} \ln \left (5\right )^{3}+36 x \ln \left (5\right )^{4}-672 x^{2} \ln \left (5\right )^{3}+1175 x^{3} \ln \left (5\right )^{2}+1008 \ln \left (5\right )^{3} x -7053 x^{2} \ln \left (5\right )^{2}+5474 x^{3} \ln \left (5\right )+10584 x \ln \left (5\right )^{2}-32886 x^{2} \ln \left (5\right )+\frac {152881 x^{3}}{16}+49392 x \ln \left (5\right )-57477 x^{2}+86445 x\) | \(109\) |
parallelrisch | \(4 x^{3} \ln \left (5\right )^{4}-24 \ln \left (5\right )^{4} x^{2}+112 x^{3} \ln \left (5\right )^{3}+36 x \ln \left (5\right )^{4}-672 x^{2} \ln \left (5\right )^{3}+1175 x^{3} \ln \left (5\right )^{2}+1008 \ln \left (5\right )^{3} x -7053 x^{2} \ln \left (5\right )^{2}+5474 x^{3} \ln \left (5\right )+10584 x \ln \left (5\right )^{2}-32886 x^{2} \ln \left (5\right )+\frac {152881 x^{3}}{16}+49392 x \ln \left (5\right )-57477 x^{2}+86445 x\) | \(109\) |
parts | \(4 x^{3} \ln \left (5\right )^{4}-24 \ln \left (5\right )^{4} x^{2}+112 x^{3} \ln \left (5\right )^{3}+36 x \ln \left (5\right )^{4}-672 x^{2} \ln \left (5\right )^{3}+1175 x^{3} \ln \left (5\right )^{2}+1008 \ln \left (5\right )^{3} x -7053 x^{2} \ln \left (5\right )^{2}+5474 x^{3} \ln \left (5\right )+10584 x \ln \left (5\right )^{2}-32886 x^{2} \ln \left (5\right )+\frac {152881 x^{3}}{16}+49392 x \ln \left (5\right )-57477 x^{2}+86445 x\) | \(109\) |
int(1/16*(192*x^2-768*x+576)*ln(5)^4+1/16*(5376*x^2-21504*x+16128)*ln(5)^3 +1/16*(56400*x^2-225696*x+169344)*ln(5)^2+1/16*(262752*x^2-1052352*x+79027 2)*ln(5)+458643/16*x^2-114954*x+86445,x,method=_RETURNVERBOSE)
(-24*ln(5)^4-672*ln(5)^3-7053*ln(5)^2-32886*ln(5)-57477)*x^2+(4*ln(5)^4+11 2*ln(5)^3+1175*ln(5)^2+5474*ln(5)+152881/16)*x^3+(36*ln(5)^4+1008*ln(5)^3+ 10584*ln(5)^2+49392*ln(5)+86445)*x
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {1}{16} \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx=4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (5\right )^{4} + 112 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (5\right )^{3} + \frac {152881}{16} \, x^{3} + {\left (1175 \, x^{3} - 7053 \, x^{2} + 10584 \, x\right )} \log \left (5\right )^{2} - 57477 \, x^{2} + 14 \, {\left (391 \, x^{3} - 2349 \, x^{2} + 3528 \, x\right )} \log \left (5\right ) + 86445 \, x \]
integrate(1/16*(192*x^2-768*x+576)*log(5)^4+1/16*(5376*x^2-21504*x+16128)* log(5)^3+1/16*(56400*x^2-225696*x+169344)*log(5)^2+1/16*(262752*x^2-105235 2*x+790272)*log(5)+458643/16*x^2-114954*x+86445,x, algorithm=\
4*(x^3 - 6*x^2 + 9*x)*log(5)^4 + 112*(x^3 - 6*x^2 + 9*x)*log(5)^3 + 152881 /16*x^3 + (1175*x^3 - 7053*x^2 + 10584*x)*log(5)^2 - 57477*x^2 + 14*(391*x ^3 - 2349*x^2 + 3528*x)*log(5) + 86445*x
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (20) = 40\).
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.41 \[ \int \frac {1}{16} \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx=x^{3} \cdot \left (4 \log {\left (5 \right )}^{4} + 112 \log {\left (5 \right )}^{3} + 1175 \log {\left (5 \right )}^{2} + 5474 \log {\left (5 \right )} + \frac {152881}{16}\right ) + x^{2} \left (-57477 - 32886 \log {\left (5 \right )} - 7053 \log {\left (5 \right )}^{2} - 672 \log {\left (5 \right )}^{3} - 24 \log {\left (5 \right )}^{4}\right ) + x \left (36 \log {\left (5 \right )}^{4} + 1008 \log {\left (5 \right )}^{3} + 10584 \log {\left (5 \right )}^{2} + 49392 \log {\left (5 \right )} + 86445\right ) \]
integrate(1/16*(192*x**2-768*x+576)*ln(5)**4+1/16*(5376*x**2-21504*x+16128 )*ln(5)**3+1/16*(56400*x**2-225696*x+169344)*ln(5)**2+1/16*(262752*x**2-10 52352*x+790272)*ln(5)+458643/16*x**2-114954*x+86445,x)
x**3*(4*log(5)**4 + 112*log(5)**3 + 1175*log(5)**2 + 5474*log(5) + 152881/ 16) + x**2*(-57477 - 32886*log(5) - 7053*log(5)**2 - 672*log(5)**3 - 24*lo g(5)**4) + x*(36*log(5)**4 + 1008*log(5)**3 + 10584*log(5)**2 + 49392*log( 5) + 86445)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).
Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {1}{16} \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx=4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (5\right )^{4} + 112 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (5\right )^{3} + \frac {152881}{16} \, x^{3} + {\left (1175 \, x^{3} - 7053 \, x^{2} + 10584 \, x\right )} \log \left (5\right )^{2} - 57477 \, x^{2} + 14 \, {\left (391 \, x^{3} - 2349 \, x^{2} + 3528 \, x\right )} \log \left (5\right ) + 86445 \, x \]
integrate(1/16*(192*x^2-768*x+576)*log(5)^4+1/16*(5376*x^2-21504*x+16128)* log(5)^3+1/16*(56400*x^2-225696*x+169344)*log(5)^2+1/16*(262752*x^2-105235 2*x+790272)*log(5)+458643/16*x^2-114954*x+86445,x, algorithm=\
4*(x^3 - 6*x^2 + 9*x)*log(5)^4 + 112*(x^3 - 6*x^2 + 9*x)*log(5)^3 + 152881 /16*x^3 + (1175*x^3 - 7053*x^2 + 10584*x)*log(5)^2 - 57477*x^2 + 14*(391*x ^3 - 2349*x^2 + 3528*x)*log(5) + 86445*x
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {1}{16} \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx=4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (5\right )^{4} + 112 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (5\right )^{3} + \frac {152881}{16} \, x^{3} + {\left (1175 \, x^{3} - 7053 \, x^{2} + 10584 \, x\right )} \log \left (5\right )^{2} - 57477 \, x^{2} + 14 \, {\left (391 \, x^{3} - 2349 \, x^{2} + 3528 \, x\right )} \log \left (5\right ) + 86445 \, x \]
integrate(1/16*(192*x^2-768*x+576)*log(5)^4+1/16*(5376*x^2-21504*x+16128)* log(5)^3+1/16*(56400*x^2-225696*x+169344)*log(5)^2+1/16*(262752*x^2-105235 2*x+790272)*log(5)+458643/16*x^2-114954*x+86445,x, algorithm=\
4*(x^3 - 6*x^2 + 9*x)*log(5)^4 + 112*(x^3 - 6*x^2 + 9*x)*log(5)^3 + 152881 /16*x^3 + (1175*x^3 - 7053*x^2 + 10584*x)*log(5)^2 - 57477*x^2 + 14*(391*x ^3 - 2349*x^2 + 3528*x)*log(5) + 86445*x
Time = 7.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.11 \[ \int \frac {1}{16} \left (1383120-1839264 x+458643 x^2+\left (790272-1052352 x+262752 x^2\right ) \log (5)+\left (169344-225696 x+56400 x^2\right ) \log ^2(5)+\left (16128-21504 x+5376 x^2\right ) \log ^3(5)+\left (576-768 x+192 x^2\right ) \log ^4(5)\right ) \, dx=\left (5474\,\ln \left (5\right )+1175\,{\ln \left (5\right )}^2+112\,{\ln \left (5\right )}^3+4\,{\ln \left (5\right )}^4+\frac {152881}{16}\right )\,x^3+\left (-32886\,\ln \left (5\right )-7053\,{\ln \left (5\right )}^2-672\,{\ln \left (5\right )}^3-24\,{\ln \left (5\right )}^4-57477\right )\,x^2+\left (49392\,\ln \left (5\right )+10584\,{\ln \left (5\right )}^2+1008\,{\ln \left (5\right )}^3+36\,{\ln \left (5\right )}^4+86445\right )\,x \]
int((log(5)*(262752*x^2 - 1052352*x + 790272))/16 - 114954*x + (log(5)^4*( 192*x^2 - 768*x + 576))/16 + (log(5)^3*(5376*x^2 - 21504*x + 16128))/16 + (log(5)^2*(56400*x^2 - 225696*x + 169344))/16 + (458643*x^2)/16 + 86445,x)