Integrand size = 22, antiderivative size = 104 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=4096 x+24576 x^2+\frac {237568 x^3}{3}+139776 x^4+\frac {538624 x^5}{5}-30720 x^6-\frac {566912 x^7}{7}+36384 x^8+\frac {641152 x^9}{9}-\frac {169584 x^{10}}{5}-\frac {331040 x^{11}}{11}+31128 x^{12}-\frac {12095 x^{13}}{13}-\frac {75504 x^{14}}{7}+\frac {102784 x^{15}}{15}-1920 x^{16}+\frac {4096 x^{17}}{17} \] Output:
4096*x+24576*x^2+237568/3*x^3+139776*x^4+538624/5*x^5-30720*x^6-566912/7*x ^7+36384*x^8+641152/9*x^9-169584/5*x^10-331040/11*x^11+31128*x^12-12095/13 *x^13-75504/7*x^14+102784/15*x^15-1920*x^16+4096/17*x^17
Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=4096 x+24576 x^2+\frac {237568 x^3}{3}+139776 x^4+\frac {538624 x^5}{5}-30720 x^6-\frac {566912 x^7}{7}+36384 x^8+\frac {641152 x^9}{9}-\frac {169584 x^{10}}{5}-\frac {331040 x^{11}}{11}+31128 x^{12}-\frac {12095 x^{13}}{13}-\frac {75504 x^{14}}{7}+\frac {102784 x^{15}}{15}-1920 x^{16}+\frac {4096 x^{17}}{17} \] Input:
Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^4,x]
Output:
4096*x + 24576*x^2 + (237568*x^3)/3 + 139776*x^4 + (538624*x^5)/5 - 30720* x^6 - (566912*x^7)/7 + 36384*x^8 + (641152*x^9)/9 - (169584*x^10)/5 - (331 040*x^11)/11 + 31128*x^12 - (12095*x^13)/13 - (75504*x^14)/7 + (102784*x^1 5)/15 - 1920*x^16 + (4096*x^17)/17
Time = 0.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2465, 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.
\(\displaystyle \int \left (8 x^4-15 x^3+8 x^2+24 x+8\right )^4 \, dx\) |
\(\Big \downarrow \) 2465 |
\(\displaystyle \int \left (4096 x^{16}-30720 x^{15}+102784 x^{14}-151008 x^{13}-12095 x^{12}+373536 x^{11}-331040 x^{10}-339168 x^9+641152 x^8+291072 x^7-566912 x^6-184320 x^5+538624 x^4+559104 x^3+237568 x^2+49152 x+4096\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4096 x^{17}}{17}-1920 x^{16}+\frac {102784 x^{15}}{15}-\frac {75504 x^{14}}{7}-\frac {12095 x^{13}}{13}+31128 x^{12}-\frac {331040 x^{11}}{11}-\frac {169584 x^{10}}{5}+\frac {641152 x^9}{9}+36384 x^8-\frac {566912 x^7}{7}-30720 x^6+\frac {538624 x^5}{5}+139776 x^4+\frac {237568 x^3}{3}+24576 x^2+4096 x\) |
Input:
Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^4,x]
Output:
4096*x + 24576*x^2 + (237568*x^3)/3 + 139776*x^4 + (538624*x^5)/5 - 30720* x^6 - (566912*x^7)/7 + 36384*x^8 + (641152*x^9)/9 - (169584*x^10)/5 - (331 040*x^11)/11 + 31128*x^12 - (12095*x^13)/13 - (75504*x^14)/7 + (102784*x^1 5)/15 - 1920*x^16 + (4096*x^17)/17
Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[ExpandToSum[u, Px^p, x], x] /; PolyQ [Px, x] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x ] && IGtQ[p, 0]
Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81
method | result | size |
orering | \(\frac {x \left (184504320 x^{16}-1470268800 x^{15}+5247225984 x^{14}-8259760080 x^{13}-712455975 x^{12}+23836732920 x^{11}-23045349600 x^{10}-25972298352 x^{9}+54552417920 x^{8}+27861593760 x^{7}-62017338240 x^{6}-23524300800 x^{5}+82491881472 x^{4}+107035568640 x^{3}+60640419840 x^{2}+18819440640 x +3136573440\right )}{765765}\) | \(84\) |
gosper | \(4096 x +24576 x^{2}+\frac {237568}{3} x^{3}+139776 x^{4}+\frac {538624}{5} x^{5}-30720 x^{6}-\frac {566912}{7} x^{7}+36384 x^{8}+\frac {641152}{9} x^{9}-\frac {169584}{5} x^{10}-\frac {331040}{11} x^{11}+31128 x^{12}-\frac {12095}{13} x^{13}-\frac {75504}{7} x^{14}+\frac {102784}{15} x^{15}-1920 x^{16}+\frac {4096}{17} x^{17}\) | \(85\) |
default | \(4096 x +24576 x^{2}+\frac {237568}{3} x^{3}+139776 x^{4}+\frac {538624}{5} x^{5}-30720 x^{6}-\frac {566912}{7} x^{7}+36384 x^{8}+\frac {641152}{9} x^{9}-\frac {169584}{5} x^{10}-\frac {331040}{11} x^{11}+31128 x^{12}-\frac {12095}{13} x^{13}-\frac {75504}{7} x^{14}+\frac {102784}{15} x^{15}-1920 x^{16}+\frac {4096}{17} x^{17}\) | \(85\) |
norman | \(4096 x +24576 x^{2}+\frac {237568}{3} x^{3}+139776 x^{4}+\frac {538624}{5} x^{5}-30720 x^{6}-\frac {566912}{7} x^{7}+36384 x^{8}+\frac {641152}{9} x^{9}-\frac {169584}{5} x^{10}-\frac {331040}{11} x^{11}+31128 x^{12}-\frac {12095}{13} x^{13}-\frac {75504}{7} x^{14}+\frac {102784}{15} x^{15}-1920 x^{16}+\frac {4096}{17} x^{17}\) | \(85\) |
risch | \(4096 x +24576 x^{2}+\frac {237568}{3} x^{3}+139776 x^{4}+\frac {538624}{5} x^{5}-30720 x^{6}-\frac {566912}{7} x^{7}+36384 x^{8}+\frac {641152}{9} x^{9}-\frac {169584}{5} x^{10}-\frac {331040}{11} x^{11}+31128 x^{12}-\frac {12095}{13} x^{13}-\frac {75504}{7} x^{14}+\frac {102784}{15} x^{15}-1920 x^{16}+\frac {4096}{17} x^{17}\) | \(85\) |
parallelrisch | \(4096 x +24576 x^{2}+\frac {237568}{3} x^{3}+139776 x^{4}+\frac {538624}{5} x^{5}-30720 x^{6}-\frac {566912}{7} x^{7}+36384 x^{8}+\frac {641152}{9} x^{9}-\frac {169584}{5} x^{10}-\frac {331040}{11} x^{11}+31128 x^{12}-\frac {12095}{13} x^{13}-\frac {75504}{7} x^{14}+\frac {102784}{15} x^{15}-1920 x^{16}+\frac {4096}{17} x^{17}\) | \(85\) |
Input:
int((8*x^4-15*x^3+8*x^2+24*x+8)^4,x,method=_RETURNVERBOSE)
Output:
1/765765*x*(184504320*x^16-1470268800*x^15+5247225984*x^14-8259760080*x^13 -712455975*x^12+23836732920*x^11-23045349600*x^10-25972298352*x^9+54552417 920*x^8+27861593760*x^7-62017338240*x^6-23524300800*x^5+82491881472*x^4+10 7035568640*x^3+60640419840*x^2+18819440640*x+3136573440)
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=\frac {4096}{17} \, x^{17} - 1920 \, x^{16} + \frac {102784}{15} \, x^{15} - \frac {75504}{7} \, x^{14} - \frac {12095}{13} \, x^{13} + 31128 \, x^{12} - \frac {331040}{11} \, x^{11} - \frac {169584}{5} \, x^{10} + \frac {641152}{9} \, x^{9} + 36384 \, x^{8} - \frac {566912}{7} \, x^{7} - 30720 \, x^{6} + \frac {538624}{5} \, x^{5} + 139776 \, x^{4} + \frac {237568}{3} \, x^{3} + 24576 \, x^{2} + 4096 \, x \] Input:
integrate((8*x^4-15*x^3+8*x^2+24*x+8)^4,x, algorithm="fricas")
Output:
4096/17*x^17 - 1920*x^16 + 102784/15*x^15 - 75504/7*x^14 - 12095/13*x^13 + 31128*x^12 - 331040/11*x^11 - 169584/5*x^10 + 641152/9*x^9 + 36384*x^8 - 566912/7*x^7 - 30720*x^6 + 538624/5*x^5 + 139776*x^4 + 237568/3*x^3 + 2457 6*x^2 + 4096*x
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=\frac {4096 x^{17}}{17} - 1920 x^{16} + \frac {102784 x^{15}}{15} - \frac {75504 x^{14}}{7} - \frac {12095 x^{13}}{13} + 31128 x^{12} - \frac {331040 x^{11}}{11} - \frac {169584 x^{10}}{5} + \frac {641152 x^{9}}{9} + 36384 x^{8} - \frac {566912 x^{7}}{7} - 30720 x^{6} + \frac {538624 x^{5}}{5} + 139776 x^{4} + \frac {237568 x^{3}}{3} + 24576 x^{2} + 4096 x \] Input:
integrate((8*x**4-15*x**3+8*x**2+24*x+8)**4,x)
Output:
4096*x**17/17 - 1920*x**16 + 102784*x**15/15 - 75504*x**14/7 - 12095*x**13 /13 + 31128*x**12 - 331040*x**11/11 - 169584*x**10/5 + 641152*x**9/9 + 363 84*x**8 - 566912*x**7/7 - 30720*x**6 + 538624*x**5/5 + 139776*x**4 + 23756 8*x**3/3 + 24576*x**2 + 4096*x
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=\frac {4096}{17} \, x^{17} - 1920 \, x^{16} + \frac {102784}{15} \, x^{15} - \frac {75504}{7} \, x^{14} - \frac {12095}{13} \, x^{13} + 31128 \, x^{12} - \frac {331040}{11} \, x^{11} - \frac {169584}{5} \, x^{10} + \frac {641152}{9} \, x^{9} + 36384 \, x^{8} - \frac {566912}{7} \, x^{7} - 30720 \, x^{6} + \frac {538624}{5} \, x^{5} + 139776 \, x^{4} + \frac {237568}{3} \, x^{3} + 24576 \, x^{2} + 4096 \, x \] Input:
integrate((8*x^4-15*x^3+8*x^2+24*x+8)^4,x, algorithm="maxima")
Output:
4096/17*x^17 - 1920*x^16 + 102784/15*x^15 - 75504/7*x^14 - 12095/13*x^13 + 31128*x^12 - 331040/11*x^11 - 169584/5*x^10 + 641152/9*x^9 + 36384*x^8 - 566912/7*x^7 - 30720*x^6 + 538624/5*x^5 + 139776*x^4 + 237568/3*x^3 + 2457 6*x^2 + 4096*x
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=\frac {4096}{17} \, x^{17} - 1920 \, x^{16} + \frac {102784}{15} \, x^{15} - \frac {75504}{7} \, x^{14} - \frac {12095}{13} \, x^{13} + 31128 \, x^{12} - \frac {331040}{11} \, x^{11} - \frac {169584}{5} \, x^{10} + \frac {641152}{9} \, x^{9} + 36384 \, x^{8} - \frac {566912}{7} \, x^{7} - 30720 \, x^{6} + \frac {538624}{5} \, x^{5} + 139776 \, x^{4} + \frac {237568}{3} \, x^{3} + 24576 \, x^{2} + 4096 \, x \] Input:
integrate((8*x^4-15*x^3+8*x^2+24*x+8)^4,x, algorithm="giac")
Output:
4096/17*x^17 - 1920*x^16 + 102784/15*x^15 - 75504/7*x^14 - 12095/13*x^13 + 31128*x^12 - 331040/11*x^11 - 169584/5*x^10 + 641152/9*x^9 + 36384*x^8 - 566912/7*x^7 - 30720*x^6 + 538624/5*x^5 + 139776*x^4 + 237568/3*x^3 + 2457 6*x^2 + 4096*x
Time = 22.95 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=\frac {4096\,x^{17}}{17}-1920\,x^{16}+\frac {102784\,x^{15}}{15}-\frac {75504\,x^{14}}{7}-\frac {12095\,x^{13}}{13}+31128\,x^{12}-\frac {331040\,x^{11}}{11}-\frac {169584\,x^{10}}{5}+\frac {641152\,x^9}{9}+36384\,x^8-\frac {566912\,x^7}{7}-30720\,x^6+\frac {538624\,x^5}{5}+139776\,x^4+\frac {237568\,x^3}{3}+24576\,x^2+4096\,x \] Input:
int((24*x + 8*x^2 - 15*x^3 + 8*x^4 + 8)^4,x)
Output:
4096*x + 24576*x^2 + (237568*x^3)/3 + 139776*x^4 + (538624*x^5)/5 - 30720* x^6 - (566912*x^7)/7 + 36384*x^8 + (641152*x^9)/9 - (169584*x^10)/5 - (331 040*x^11)/11 + 31128*x^12 - (12095*x^13)/13 - (75504*x^14)/7 + (102784*x^1 5)/15 - 1920*x^16 + (4096*x^17)/17
Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.80 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx=\frac {x \left (184504320 x^{16}-1470268800 x^{15}+5247225984 x^{14}-8259760080 x^{13}-712455975 x^{12}+23836732920 x^{11}-23045349600 x^{10}-25972298352 x^{9}+54552417920 x^{8}+27861593760 x^{7}-62017338240 x^{6}-23524300800 x^{5}+82491881472 x^{4}+107035568640 x^{3}+60640419840 x^{2}+18819440640 x +3136573440\right )}{765765} \] Input:
int((8*x^4-15*x^3+8*x^2+24*x+8)^4,x)
Output:
(x*(184504320*x**16 - 1470268800*x**15 + 5247225984*x**14 - 8259760080*x** 13 - 712455975*x**12 + 23836732920*x**11 - 23045349600*x**10 - 25972298352 *x**9 + 54552417920*x**8 + 27861593760*x**7 - 62017338240*x**6 - 235243008 00*x**5 + 82491881472*x**4 + 107035568640*x**3 + 60640419840*x**2 + 188194 40640*x + 3136573440))/765765