Integrand size = 82, antiderivative size = 21 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=(2+x-(-1+x) x (x+(-3+5 x) \log (x)))^2 \] Output:
(2-x*(-1+x)*(ln(x)*(5*x-3)+x)+x)^2
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(21)=42\).
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.33 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=4 x+5 x^2-2 x^3-x^4-2 x^5+x^6-12 x \log (x)+26 x^2 \log (x)-10 x^3 \log (x)+12 x^4 \log (x)-26 x^5 \log (x)+10 x^6 \log (x)+9 x^2 \log ^2(x)-48 x^3 \log ^2(x)+94 x^4 \log ^2(x)-80 x^5 \log ^2(x)+25 x^6 \log ^2(x) \] Input:
Integrate[-8 + 36*x - 16*x^2 + 8*x^3 - 36*x^4 + 16*x^5 + (-12 + 70*x - 126 *x^2 + 236*x^3 - 290*x^4 + 110*x^5)*Log[x] + (18*x - 144*x^2 + 376*x^3 - 4 00*x^4 + 150*x^5)*Log[x]^2,x]
Output:
4*x + 5*x^2 - 2*x^3 - x^4 - 2*x^5 + x^6 - 12*x*Log[x] + 26*x^2*Log[x] - 10 *x^3*Log[x] + 12*x^4*Log[x] - 26*x^5*Log[x] + 10*x^6*Log[x] + 9*x^2*Log[x] ^2 - 48*x^3*Log[x]^2 + 94*x^4*Log[x]^2 - 80*x^5*Log[x]^2 + 25*x^6*Log[x]^2
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(21)=42\).
Time = 0.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.33, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {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.
\(\displaystyle \int \left (16 x^5-36 x^4+8 x^3-16 x^2+\left (150 x^5-400 x^4+376 x^3-144 x^2+18 x\right ) \log ^2(x)+\left (110 x^5-290 x^4+236 x^3-126 x^2+70 x-12\right ) \log (x)+36 x-8\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^6+25 x^6 \log ^2(x)+10 x^6 \log (x)-2 x^5-80 x^5 \log ^2(x)-26 x^5 \log (x)-x^4+94 x^4 \log ^2(x)+12 x^4 \log (x)-2 x^3-48 x^3 \log ^2(x)-10 x^3 \log (x)+5 x^2+9 x^2 \log ^2(x)+26 x^2 \log (x)+4 x-12 x \log (x)\) |
Input:
Int[-8 + 36*x - 16*x^2 + 8*x^3 - 36*x^4 + 16*x^5 + (-12 + 70*x - 126*x^2 + 236*x^3 - 290*x^4 + 110*x^5)*Log[x] + (18*x - 144*x^2 + 376*x^3 - 400*x^4 + 150*x^5)*Log[x]^2,x]
Output:
4*x + 5*x^2 - 2*x^3 - x^4 - 2*x^5 + x^6 - 12*x*Log[x] + 26*x^2*Log[x] - 10 *x^3*Log[x] + 12*x^4*Log[x] - 26*x^5*Log[x] + 10*x^6*Log[x] + 9*x^2*Log[x] ^2 - 48*x^3*Log[x]^2 + 94*x^4*Log[x]^2 - 80*x^5*Log[x]^2 + 25*x^6*Log[x]^2
Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(21)=42\).
Time = 1.68 (sec) , antiderivative size = 113, normalized size of antiderivative = 5.38
method | result | size |
default | \(4 x +12 x^{4} \ln \left (x \right )+94 x^{4} \ln \left (x \right )^{2}+10 x^{6} \ln \left (x \right )+25 x^{6} \ln \left (x \right )^{2}-80 x^{5} \ln \left (x \right )^{2}-48 x^{3} \ln \left (x \right )^{2}-10 x^{3} \ln \left (x \right )+9 x^{2} \ln \left (x \right )^{2}-26 x^{5} \ln \left (x \right )+26 x^{2} \ln \left (x \right )-12 x \ln \left (x \right )-2 x^{5}+x^{6}+5 x^{2}-2 x^{3}-x^{4}\) | \(113\) |
norman | \(4 x +12 x^{4} \ln \left (x \right )+94 x^{4} \ln \left (x \right )^{2}+10 x^{6} \ln \left (x \right )+25 x^{6} \ln \left (x \right )^{2}-80 x^{5} \ln \left (x \right )^{2}-48 x^{3} \ln \left (x \right )^{2}-10 x^{3} \ln \left (x \right )+9 x^{2} \ln \left (x \right )^{2}-26 x^{5} \ln \left (x \right )+26 x^{2} \ln \left (x \right )-12 x \ln \left (x \right )-2 x^{5}+x^{6}+5 x^{2}-2 x^{3}-x^{4}\) | \(113\) |
parallelrisch | \(4 x +12 x^{4} \ln \left (x \right )+94 x^{4} \ln \left (x \right )^{2}+10 x^{6} \ln \left (x \right )+25 x^{6} \ln \left (x \right )^{2}-80 x^{5} \ln \left (x \right )^{2}-48 x^{3} \ln \left (x \right )^{2}-10 x^{3} \ln \left (x \right )+9 x^{2} \ln \left (x \right )^{2}-26 x^{5} \ln \left (x \right )+26 x^{2} \ln \left (x \right )-12 x \ln \left (x \right )-2 x^{5}+x^{6}+5 x^{2}-2 x^{3}-x^{4}\) | \(113\) |
parts | \(4 x +12 x^{4} \ln \left (x \right )+94 x^{4} \ln \left (x \right )^{2}+10 x^{6} \ln \left (x \right )+25 x^{6} \ln \left (x \right )^{2}-80 x^{5} \ln \left (x \right )^{2}-48 x^{3} \ln \left (x \right )^{2}-10 x^{3} \ln \left (x \right )+9 x^{2} \ln \left (x \right )^{2}-26 x^{5} \ln \left (x \right )+26 x^{2} \ln \left (x \right )-12 x \ln \left (x \right )-2 x^{5}+x^{6}+5 x^{2}-2 x^{3}-x^{4}\) | \(113\) |
risch | \(\left (25 x^{6}-80 x^{5}+94 x^{4}-48 x^{3}+9 x^{2}\right ) \ln \left (x \right )^{2}+\left (-\frac {25}{3} x^{6}+32 x^{5}-47 x^{4}+32 x^{3}-9 x^{2}\right ) \ln \left (x \right )+x^{6}-2 x^{5}-x^{4}-2 x^{3}+5 x^{2}+\left (\frac {55}{3} x^{6}-58 x^{5}+59 x^{4}-42 x^{3}+35 x^{2}-12 x \right ) \ln \left (x \right )+4 x\) | \(120\) |
orering | \(\frac {\left (7998046875 x^{14}-69176953125 x^{13}+276328406250 x^{12}-649712840625 x^{11}+962089588125 x^{10}-879465048750 x^{9}+129209996250 x^{8}+1209194158750 x^{7}-2230081862675 x^{6}+1969809462115 x^{5}-922162539714 x^{4}+207736714203 x^{3}-14802108789 x^{2}-85202136 x -159015474\right ) x \left (\left (150 x^{5}-400 x^{4}+376 x^{3}-144 x^{2}+18 x \right ) \ln \left (x \right )^{2}+\left (110 x^{5}-290 x^{4}+236 x^{3}-126 x^{2}+70 x -12\right ) \ln \left (x \right )+16 x^{5}-36 x^{4}+8 x^{3}-16 x^{2}+36 x -8\right )}{18984375000 x^{14}-152163281250 x^{13}+561018093750 x^{12}-1217359631250 x^{11}+1674967117500 x^{10}-1461215880000 x^{9}+733528620000 x^{8}-110672167500 x^{7}-98262112500 x^{6}+66161193750 x^{5}-15950058750 x^{4}+756168750 x^{3}+375840000 x^{2}-69862500 x +3645000}-\frac {x^{2} \left (1318359375 x^{14}-12255468750 x^{13}+52863890625 x^{12}-134707185000 x^{11}+216487940625 x^{10}-213364293750 x^{9}+508223272000 x^{7}-1049022855195 x^{6}+1087000472130 x^{5}-604383456355 x^{4}+155808071088 x^{3}-3603290925 x^{2}-5564020230 x +706684554\right ) \left (\left (750 x^{4}-1600 x^{3}+1128 x^{2}-288 x +18\right ) \ln \left (x \right )^{2}+\frac {2 \left (150 x^{5}-400 x^{4}+376 x^{3}-144 x^{2}+18 x \right ) \ln \left (x \right )}{x}+\left (550 x^{4}-1160 x^{3}+708 x^{2}-252 x +70\right ) \ln \left (x \right )+\frac {110 x^{5}-290 x^{4}+236 x^{3}-126 x^{2}+70 x -12}{x}+80 x^{4}-144 x^{3}+24 x^{2}-32 x +36\right )}{11250 \left (1687500 x^{14}-13525625 x^{13}+49868275 x^{12}-108209745 x^{11}+148885966 x^{10}-129885856 x^{9}+65202544 x^{8}-9837526 x^{7}-8734410 x^{6}+5880995 x^{5}-1417783 x^{4}+67215 x^{3}+33408 x^{2}-6210 x +324\right )}+\frac {x^{2} \left (87890625 x^{15}-875390625 x^{14}+4066453125 x^{13}-11225598750 x^{12}+19680721875 x^{11}-21336429375 x^{10}+63527909000 x^{8}-149860407885 x^{7}+181166745355 x^{6}-120876691271 x^{5}+38952017772 x^{4}-1201096975 x^{3}-2782010115 x^{2}+706684554 x -54220158\right ) \left (\left (3000 x^{3}-4800 x^{2}+2256 x -288\right ) \ln \left (x \right )^{2}+\frac {4 \left (750 x^{4}-1600 x^{3}+1128 x^{2}-288 x +18\right ) \ln \left (x \right )}{x}+\frac {300 x^{5}-800 x^{4}+752 x^{3}-288 x^{2}+36 x}{x^{2}}-\frac {2 \left (150 x^{5}-400 x^{4}+376 x^{3}-144 x^{2}+18 x \right ) \ln \left (x \right )}{x^{2}}+\left (2200 x^{3}-3480 x^{2}+1416 x -252\right ) \ln \left (x \right )+\frac {1100 x^{4}-2320 x^{3}+1416 x^{2}-504 x +140}{x}-\frac {110 x^{5}-290 x^{4}+236 x^{3}-126 x^{2}+70 x -12}{x^{2}}+320 x^{3}-432 x^{2}+48 x -32\right )}{11250 \left (25 x^{5}-83 x^{4}+110 x^{3}-29 x^{2}-23 x +6\right ) \left (67500 x^{9}-316925 x^{8}+645540 x^{7}-712427 x^{6}+444272 x^{5}-144717 x^{4}+16388 x^{3}+2655 x^{2}-828 x +54\right )}\) | \(843\) |
Input:
int((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*ln(x)^2+(110*x^5-290*x^4+236*x^ 3-126*x^2+70*x-12)*ln(x)+16*x^5-36*x^4+8*x^3-16*x^2+36*x-8,x,method=_RETUR NVERBOSE)
Output:
4*x+12*x^4*ln(x)+94*x^4*ln(x)^2+10*x^6*ln(x)+25*x^6*ln(x)^2-80*x^5*ln(x)^2 -48*x^3*ln(x)^2-10*x^3*ln(x)+9*x^2*ln(x)^2-26*x^5*ln(x)+26*x^2*ln(x)-12*x* ln(x)-2*x^5+x^6+5*x^2-2*x^3-x^4
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.33 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=x^{6} - 2 \, x^{5} - x^{4} - 2 \, x^{3} + {\left (25 \, x^{6} - 80 \, x^{5} + 94 \, x^{4} - 48 \, x^{3} + 9 \, x^{2}\right )} \log \left (x\right )^{2} + 5 \, x^{2} + 2 \, {\left (5 \, x^{6} - 13 \, x^{5} + 6 \, x^{4} - 5 \, x^{3} + 13 \, x^{2} - 6 \, x\right )} \log \left (x\right ) + 4 \, x \] Input:
integrate((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*log(x)^2+(110*x^5-290*x^4 +236*x^3-126*x^2+70*x-12)*log(x)+16*x^5-36*x^4+8*x^3-16*x^2+36*x-8,x, algo rithm="fricas")
Output:
x^6 - 2*x^5 - x^4 - 2*x^3 + (25*x^6 - 80*x^5 + 94*x^4 - 48*x^3 + 9*x^2)*lo g(x)^2 + 5*x^2 + 2*(5*x^6 - 13*x^5 + 6*x^4 - 5*x^3 + 13*x^2 - 6*x)*log(x) + 4*x
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.14 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=x^{6} - 2 x^{5} - x^{4} - 2 x^{3} + 5 x^{2} + 4 x + \left (25 x^{6} - 80 x^{5} + 94 x^{4} - 48 x^{3} + 9 x^{2}\right ) \log {\left (x \right )}^{2} + \left (10 x^{6} - 26 x^{5} + 12 x^{4} - 10 x^{3} + 26 x^{2} - 12 x\right ) \log {\left (x \right )} \] Input:
integrate((150*x**5-400*x**4+376*x**3-144*x**2+18*x)*ln(x)**2+(110*x**5-29 0*x**4+236*x**3-126*x**2+70*x-12)*ln(x)+16*x**5-36*x**4+8*x**3-16*x**2+36* x-8,x)
Output:
x**6 - 2*x**5 - x**4 - 2*x**3 + 5*x**2 + 4*x + (25*x**6 - 80*x**5 + 94*x** 4 - 48*x**3 + 9*x**2)*log(x)**2 + (10*x**6 - 26*x**5 + 12*x**4 - 10*x**3 + 26*x**2 - 12*x)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (22) = 44\).
Time = 0.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 7.00 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=\frac {25}{18} \, {\left (18 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 1\right )} x^{6} - \frac {16}{5} \, {\left (25 \, \log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 2\right )} x^{5} - \frac {7}{18} \, x^{6} + \frac {47}{4} \, {\left (8 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} x^{4} + \frac {22}{5} \, x^{5} - \frac {16}{3} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} - \frac {51}{4} \, x^{4} + \frac {9}{2} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} + \frac {26}{3} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {1}{3} \, {\left (55 \, x^{6} - 174 \, x^{5} + 177 \, x^{4} - 126 \, x^{3} + 105 \, x^{2} - 36 \, x\right )} \log \left (x\right ) + 4 \, x \] Input:
integrate((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*log(x)^2+(110*x^5-290*x^4 +236*x^3-126*x^2+70*x-12)*log(x)+16*x^5-36*x^4+8*x^3-16*x^2+36*x-8,x, algo rithm="maxima")
Output:
25/18*(18*log(x)^2 - 6*log(x) + 1)*x^6 - 16/5*(25*log(x)^2 - 10*log(x) + 2 )*x^5 - 7/18*x^6 + 47/4*(8*log(x)^2 - 4*log(x) + 1)*x^4 + 22/5*x^5 - 16/3* (9*log(x)^2 - 6*log(x) + 2)*x^3 - 51/4*x^4 + 9/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 26/3*x^3 + 1/2*x^2 + 1/3*(55*x^6 - 174*x^5 + 177*x^4 - 126*x^3 + 105*x^2 - 36*x)*log(x) + 4*x
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.33 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=25 \, x^{6} \log \left (x\right )^{2} + 10 \, x^{6} \log \left (x\right ) - 80 \, x^{5} \log \left (x\right )^{2} + x^{6} - 26 \, x^{5} \log \left (x\right ) + 94 \, x^{4} \log \left (x\right )^{2} - 2 \, x^{5} + 12 \, x^{4} \log \left (x\right ) - 48 \, x^{3} \log \left (x\right )^{2} - x^{4} - 10 \, x^{3} \log \left (x\right ) + 9 \, x^{2} \log \left (x\right )^{2} - 2 \, x^{3} + 26 \, x^{2} \log \left (x\right ) + 5 \, x^{2} - 12 \, x \log \left (x\right ) + 4 \, x \] Input:
integrate((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*log(x)^2+(110*x^5-290*x^4 +236*x^3-126*x^2+70*x-12)*log(x)+16*x^5-36*x^4+8*x^3-16*x^2+36*x-8,x, algo rithm="giac")
Output:
25*x^6*log(x)^2 + 10*x^6*log(x) - 80*x^5*log(x)^2 + x^6 - 26*x^5*log(x) + 94*x^4*log(x)^2 - 2*x^5 + 12*x^4*log(x) - 48*x^3*log(x)^2 - x^4 - 10*x^3*l og(x) + 9*x^2*log(x)^2 - 2*x^3 + 26*x^2*log(x) + 5*x^2 - 12*x*log(x) + 4*x
Time = 3.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.33 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=25\,x^6\,{\ln \left (x\right )}^2+10\,x^6\,\ln \left (x\right )+x^6-80\,x^5\,{\ln \left (x\right )}^2-26\,x^5\,\ln \left (x\right )-2\,x^5+94\,x^4\,{\ln \left (x\right )}^2+12\,x^4\,\ln \left (x\right )-x^4-48\,x^3\,{\ln \left (x\right )}^2-10\,x^3\,\ln \left (x\right )-2\,x^3+9\,x^2\,{\ln \left (x\right )}^2+26\,x^2\,\ln \left (x\right )+5\,x^2-12\,x\,\ln \left (x\right )+4\,x \] Input:
int(36*x + log(x)*(70*x - 126*x^2 + 236*x^3 - 290*x^4 + 110*x^5 - 12) + lo g(x)^2*(18*x - 144*x^2 + 376*x^3 - 400*x^4 + 150*x^5) - 16*x^2 + 8*x^3 - 3 6*x^4 + 16*x^5 - 8,x)
Output:
4*x + 26*x^2*log(x) - 10*x^3*log(x) + 12*x^4*log(x) - 26*x^5*log(x) + 10*x ^6*log(x) + 9*x^2*log(x)^2 - 48*x^3*log(x)^2 + 94*x^4*log(x)^2 - 80*x^5*lo g(x)^2 + 25*x^6*log(x)^2 - 12*x*log(x) + 5*x^2 - 2*x^3 - x^4 - 2*x^5 + x^6
Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.00 \[ \int \left (-8+36 x-16 x^2+8 x^3-36 x^4+16 x^5+\left (-12+70 x-126 x^2+236 x^3-290 x^4+110 x^5\right ) \log (x)+\left (18 x-144 x^2+376 x^3-400 x^4+150 x^5\right ) \log ^2(x)\right ) \, dx=x \left (25 \mathrm {log}\left (x \right )^{2} x^{5}-80 \mathrm {log}\left (x \right )^{2} x^{4}+94 \mathrm {log}\left (x \right )^{2} x^{3}-48 \mathrm {log}\left (x \right )^{2} x^{2}+9 \mathrm {log}\left (x \right )^{2} x +10 \,\mathrm {log}\left (x \right ) x^{5}-26 \,\mathrm {log}\left (x \right ) x^{4}+12 \,\mathrm {log}\left (x \right ) x^{3}-10 \,\mathrm {log}\left (x \right ) x^{2}+26 \,\mathrm {log}\left (x \right ) x -12 \,\mathrm {log}\left (x \right )+x^{5}-2 x^{4}-x^{3}-2 x^{2}+5 x +4\right ) \] Input:
int((150*x^5-400*x^4+376*x^3-144*x^2+18*x)*log(x)^2+(110*x^5-290*x^4+236*x ^3-126*x^2+70*x-12)*log(x)+16*x^5-36*x^4+8*x^3-16*x^2+36*x-8,x)
Output:
x*(25*log(x)**2*x**5 - 80*log(x)**2*x**4 + 94*log(x)**2*x**3 - 48*log(x)** 2*x**2 + 9*log(x)**2*x + 10*log(x)*x**5 - 26*log(x)*x**4 + 12*log(x)*x**3 - 10*log(x)*x**2 + 26*log(x)*x - 12*log(x) + x**5 - 2*x**4 - x**3 - 2*x**2 + 5*x + 4)