Integrand size = 17, antiderivative size = 97 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=x+8 x^2+\frac {112 x^3}{3}+112 x^4+\frac {1136 x^5}{5}+\frac {992 x^6}{3}+\frac {2752 x^7}{7}+448 x^8+\frac {4192 x^9}{9}+384 x^{10}+\frac {3328 x^{11}}{11}+256 x^{12}+\frac {1792 x^{13}}{13}+\frac {512 x^{14}}{7}+\frac {1024 x^{15}}{15}+\frac {256 x^{17}}{17} \] Output:
x+8*x^2+112/3*x^3+112*x^4+1136/5*x^5+992/3*x^6+2752/7*x^7+448*x^8+4192/9*x ^9+384*x^10+3328/11*x^11+256*x^12+1792/13*x^13+512/7*x^14+1024/15*x^15+256 /17*x^17
Time = 0.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=x+8 x^2+\frac {112 x^3}{3}+112 x^4+\frac {1136 x^5}{5}+\frac {992 x^6}{3}+\frac {2752 x^7}{7}+448 x^8+\frac {4192 x^9}{9}+384 x^{10}+\frac {3328 x^{11}}{11}+256 x^{12}+\frac {1792 x^{13}}{13}+\frac {512 x^{14}}{7}+\frac {1024 x^{15}}{15}+\frac {256 x^{17}}{17} \] Input:
Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^4,x]
Output:
x + 8*x^2 + (112*x^3)/3 + 112*x^4 + (1136*x^5)/5 + (992*x^6)/3 + (2752*x^7 )/7 + 448*x^8 + (4192*x^9)/9 + 384*x^10 + (3328*x^11)/11 + 256*x^12 + (179 2*x^13)/13 + (512*x^14)/7 + (1024*x^15)/15 + (256*x^17)/17
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
\(\displaystyle \int \left (4 x^4+4 x^2+4 x+1\right )^4 \, dx\) |
\(\Big \downarrow \) 2465 |
\(\displaystyle \int \left (256 x^{16}+1024 x^{14}+1024 x^{13}+1792 x^{12}+3072 x^{11}+3328 x^{10}+3840 x^9+4192 x^8+3584 x^7+2752 x^6+1984 x^5+1136 x^4+448 x^3+112 x^2+16 x+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {256 x^{17}}{17}+\frac {1024 x^{15}}{15}+\frac {512 x^{14}}{7}+\frac {1792 x^{13}}{13}+256 x^{12}+\frac {3328 x^{11}}{11}+384 x^{10}+\frac {4192 x^9}{9}+448 x^8+\frac {2752 x^7}{7}+\frac {992 x^6}{3}+\frac {1136 x^5}{5}+112 x^4+\frac {112 x^3}{3}+8 x^2+x\) |
Input:
Int[(1 + 4*x + 4*x^2 + 4*x^4)^4,x]
Output:
x + 8*x^2 + (112*x^3)/3 + 112*x^4 + (1136*x^5)/5 + (992*x^6)/3 + (2752*x^7 )/7 + 448*x^8 + (4192*x^9)/9 + 384*x^10 + (3328*x^11)/11 + 256*x^12 + (179 2*x^13)/13 + (512*x^14)/7 + (1024*x^15)/15 + (256*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 = 78, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(x +8 x^{2}+\frac {112}{3} x^{3}+112 x^{4}+\frac {1136}{5} x^{5}+\frac {992}{3} x^{6}+\frac {2752}{7} x^{7}+448 x^{8}+\frac {4192}{9} x^{9}+384 x^{10}+\frac {3328}{11} x^{11}+256 x^{12}+\frac {1792}{13} x^{13}+\frac {512}{7} x^{14}+\frac {1024}{15} x^{15}+\frac {256}{17} x^{17}\) | \(78\) |
default | \(x +8 x^{2}+\frac {112}{3} x^{3}+112 x^{4}+\frac {1136}{5} x^{5}+\frac {992}{3} x^{6}+\frac {2752}{7} x^{7}+448 x^{8}+\frac {4192}{9} x^{9}+384 x^{10}+\frac {3328}{11} x^{11}+256 x^{12}+\frac {1792}{13} x^{13}+\frac {512}{7} x^{14}+\frac {1024}{15} x^{15}+\frac {256}{17} x^{17}\) | \(78\) |
norman | \(x +8 x^{2}+\frac {112}{3} x^{3}+112 x^{4}+\frac {1136}{5} x^{5}+\frac {992}{3} x^{6}+\frac {2752}{7} x^{7}+448 x^{8}+\frac {4192}{9} x^{9}+384 x^{10}+\frac {3328}{11} x^{11}+256 x^{12}+\frac {1792}{13} x^{13}+\frac {512}{7} x^{14}+\frac {1024}{15} x^{15}+\frac {256}{17} x^{17}\) | \(78\) |
risch | \(x +8 x^{2}+\frac {112}{3} x^{3}+112 x^{4}+\frac {1136}{5} x^{5}+\frac {992}{3} x^{6}+\frac {2752}{7} x^{7}+448 x^{8}+\frac {4192}{9} x^{9}+384 x^{10}+\frac {3328}{11} x^{11}+256 x^{12}+\frac {1792}{13} x^{13}+\frac {512}{7} x^{14}+\frac {1024}{15} x^{15}+\frac {256}{17} x^{17}\) | \(78\) |
parallelrisch | \(x +8 x^{2}+\frac {112}{3} x^{3}+112 x^{4}+\frac {1136}{5} x^{5}+\frac {992}{3} x^{6}+\frac {2752}{7} x^{7}+448 x^{8}+\frac {4192}{9} x^{9}+384 x^{10}+\frac {3328}{11} x^{11}+256 x^{12}+\frac {1792}{13} x^{13}+\frac {512}{7} x^{14}+\frac {1024}{15} x^{15}+\frac {256}{17} x^{17}\) | \(78\) |
orering | \(\frac {x \left (11531520 x^{16}+52276224 x^{14}+56010240 x^{13}+105557760 x^{12}+196035840 x^{11}+231678720 x^{10}+294053760 x^{9}+356676320 x^{8}+343062720 x^{7}+301055040 x^{6}+253212960 x^{5}+173981808 x^{4}+85765680 x^{3}+28588560 x^{2}+6126120 x +765765\right )}{765765}\) | \(79\) |
Input:
int((4*x^4+4*x^2+4*x+1)^4,x,method=_RETURNVERBOSE)
Output:
x+8*x^2+112/3*x^3+112*x^4+1136/5*x^5+992/3*x^6+2752/7*x^7+448*x^8+4192/9*x ^9+384*x^10+3328/11*x^11+256*x^12+1792/13*x^13+512/7*x^14+1024/15*x^15+256 /17*x^17
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256}{17} \, x^{17} + \frac {1024}{15} \, x^{15} + \frac {512}{7} \, x^{14} + \frac {1792}{13} \, x^{13} + 256 \, x^{12} + \frac {3328}{11} \, x^{11} + 384 \, x^{10} + \frac {4192}{9} \, x^{9} + 448 \, x^{8} + \frac {2752}{7} \, x^{7} + \frac {992}{3} \, x^{6} + \frac {1136}{5} \, x^{5} + 112 \, x^{4} + \frac {112}{3} \, x^{3} + 8 \, x^{2} + x \] Input:
integrate((4*x^4+4*x^2+4*x+1)^4,x, algorithm="fricas")
Output:
256/17*x^17 + 1024/15*x^15 + 512/7*x^14 + 1792/13*x^13 + 256*x^12 + 3328/1 1*x^11 + 384*x^10 + 4192/9*x^9 + 448*x^8 + 2752/7*x^7 + 992/3*x^6 + 1136/5 *x^5 + 112*x^4 + 112/3*x^3 + 8*x^2 + x
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256 x^{17}}{17} + \frac {1024 x^{15}}{15} + \frac {512 x^{14}}{7} + \frac {1792 x^{13}}{13} + 256 x^{12} + \frac {3328 x^{11}}{11} + 384 x^{10} + \frac {4192 x^{9}}{9} + 448 x^{8} + \frac {2752 x^{7}}{7} + \frac {992 x^{6}}{3} + \frac {1136 x^{5}}{5} + 112 x^{4} + \frac {112 x^{3}}{3} + 8 x^{2} + x \] Input:
integrate((4*x**4+4*x**2+4*x+1)**4,x)
Output:
256*x**17/17 + 1024*x**15/15 + 512*x**14/7 + 1792*x**13/13 + 256*x**12 + 3 328*x**11/11 + 384*x**10 + 4192*x**9/9 + 448*x**8 + 2752*x**7/7 + 992*x**6 /3 + 1136*x**5/5 + 112*x**4 + 112*x**3/3 + 8*x**2 + x
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256}{17} \, x^{17} + \frac {1024}{15} \, x^{15} + \frac {512}{7} \, x^{14} + \frac {1792}{13} \, x^{13} + 256 \, x^{12} + \frac {3328}{11} \, x^{11} + 384 \, x^{10} + \frac {4192}{9} \, x^{9} + 448 \, x^{8} + \frac {2752}{7} \, x^{7} + \frac {992}{3} \, x^{6} + \frac {1136}{5} \, x^{5} + 112 \, x^{4} + \frac {112}{3} \, x^{3} + 8 \, x^{2} + x \] Input:
integrate((4*x^4+4*x^2+4*x+1)^4,x, algorithm="maxima")
Output:
256/17*x^17 + 1024/15*x^15 + 512/7*x^14 + 1792/13*x^13 + 256*x^12 + 3328/1 1*x^11 + 384*x^10 + 4192/9*x^9 + 448*x^8 + 2752/7*x^7 + 992/3*x^6 + 1136/5 *x^5 + 112*x^4 + 112/3*x^3 + 8*x^2 + x
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256}{17} \, x^{17} + \frac {1024}{15} \, x^{15} + \frac {512}{7} \, x^{14} + \frac {1792}{13} \, x^{13} + 256 \, x^{12} + \frac {3328}{11} \, x^{11} + 384 \, x^{10} + \frac {4192}{9} \, x^{9} + 448 \, x^{8} + \frac {2752}{7} \, x^{7} + \frac {992}{3} \, x^{6} + \frac {1136}{5} \, x^{5} + 112 \, x^{4} + \frac {112}{3} \, x^{3} + 8 \, x^{2} + x \] Input:
integrate((4*x^4+4*x^2+4*x+1)^4,x, algorithm="giac")
Output:
256/17*x^17 + 1024/15*x^15 + 512/7*x^14 + 1792/13*x^13 + 256*x^12 + 3328/1 1*x^11 + 384*x^10 + 4192/9*x^9 + 448*x^8 + 2752/7*x^7 + 992/3*x^6 + 1136/5 *x^5 + 112*x^4 + 112/3*x^3 + 8*x^2 + x
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256\,x^{17}}{17}+\frac {1024\,x^{15}}{15}+\frac {512\,x^{14}}{7}+\frac {1792\,x^{13}}{13}+256\,x^{12}+\frac {3328\,x^{11}}{11}+384\,x^{10}+\frac {4192\,x^9}{9}+448\,x^8+\frac {2752\,x^7}{7}+\frac {992\,x^6}{3}+\frac {1136\,x^5}{5}+112\,x^4+\frac {112\,x^3}{3}+8\,x^2+x \] Input:
int((4*x + 4*x^2 + 4*x^4 + 1)^4,x)
Output:
x + 8*x^2 + (112*x^3)/3 + 112*x^4 + (1136*x^5)/5 + (992*x^6)/3 + (2752*x^7 )/7 + 448*x^8 + (4192*x^9)/9 + 384*x^10 + (3328*x^11)/11 + 256*x^12 + (179 2*x^13)/13 + (512*x^14)/7 + (1024*x^15)/15 + (256*x^17)/17
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {x \left (11531520 x^{16}+52276224 x^{14}+56010240 x^{13}+105557760 x^{12}+196035840 x^{11}+231678720 x^{10}+294053760 x^{9}+356676320 x^{8}+343062720 x^{7}+301055040 x^{6}+253212960 x^{5}+173981808 x^{4}+85765680 x^{3}+28588560 x^{2}+6126120 x +765765\right )}{765765} \] Input:
int((4*x^4+4*x^2+4*x+1)^4,x)
Output:
(x*(11531520*x**16 + 52276224*x**14 + 56010240*x**13 + 105557760*x**12 + 1 96035840*x**11 + 231678720*x**10 + 294053760*x**9 + 356676320*x**8 + 34306 2720*x**7 + 301055040*x**6 + 253212960*x**5 + 173981808*x**4 + 85765680*x* *3 + 28588560*x**2 + 6126120*x + 765765))/765765