Integrand size = 17, antiderivative size = 80 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{12} (1+x)^{12}-\frac {8}{13} (1+x)^{13}+2 (1+x)^{14}-\frac {56}{15} (1+x)^{15}+\frac {35}{8} (1+x)^{16}-\frac {56}{17} (1+x)^{17}+\frac {14}{9} (1+x)^{18}-\frac {8}{19} (1+x)^{19}+\frac {1}{20} (1+x)^{20} \]
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Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 45} \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{20} (x+1)^{20}-\frac {8}{19} (x+1)^{19}+\frac {14}{9} (x+1)^{18}-\frac {56}{17} (x+1)^{17}+\frac {35}{8} (x+1)^{16}-\frac {56}{15} (x+1)^{15}+2 (x+1)^{14}-\frac {8}{13} (x+1)^{13}+\frac {1}{12} (x+1)^{12} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int x^8 (1+x)^{11} \, dx \\ & = \int \left ((1+x)^{11}-8 (1+x)^{12}+28 (1+x)^{13}-56 (1+x)^{14}+70 (1+x)^{15}-56 (1+x)^{16}+28 (1+x)^{17}-8 (1+x)^{18}+(1+x)^{19}\right ) \, dx \\ & = \frac {1}{12} (1+x)^{12}-\frac {8}{13} (1+x)^{13}+2 (1+x)^{14}-\frac {56}{15} (1+x)^{15}+\frac {35}{8} (1+x)^{16}-\frac {56}{17} (1+x)^{17}+\frac {14}{9} (1+x)^{18}-\frac {8}{19} (1+x)^{19}+\frac {1}{20} (1+x)^{20} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^9}{9}+\frac {11 x^{10}}{10}+5 x^{11}+\frac {55 x^{12}}{4}+\frac {330 x^{13}}{13}+33 x^{14}+\frac {154 x^{15}}{5}+\frac {165 x^{16}}{8}+\frac {165 x^{17}}{17}+\frac {55 x^{18}}{18}+\frac {11 x^{19}}{19}+\frac {x^{20}}{20} \]
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Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76
method | result | size |
gosper | \(\frac {x^{9} \left (75582 x^{11}+875160 x^{10}+4618900 x^{9}+14671800 x^{8}+31177575 x^{7}+46558512 x^{6}+49884120 x^{5}+38372400 x^{4}+20785050 x^{3}+7558200 x^{2}+1662804 x +167960\right )}{1511640}\) | \(61\) |
default | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
norman | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
risch | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
parallelrisch | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
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Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{20} \, x^{20} + \frac {11}{19} \, x^{19} + \frac {55}{18} \, x^{18} + \frac {165}{17} \, x^{17} + \frac {165}{8} \, x^{16} + \frac {154}{5} \, x^{15} + 33 \, x^{14} + \frac {330}{13} \, x^{13} + \frac {55}{4} \, x^{12} + 5 \, x^{11} + \frac {11}{10} \, x^{10} + \frac {1}{9} \, x^{9} \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{20}}{20} + \frac {11 x^{19}}{19} + \frac {55 x^{18}}{18} + \frac {165 x^{17}}{17} + \frac {165 x^{16}}{8} + \frac {154 x^{15}}{5} + 33 x^{14} + \frac {330 x^{13}}{13} + \frac {55 x^{12}}{4} + 5 x^{11} + \frac {11 x^{10}}{10} + \frac {x^{9}}{9} \]
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Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{20} \, x^{20} + \frac {11}{19} \, x^{19} + \frac {55}{18} \, x^{18} + \frac {165}{17} \, x^{17} + \frac {165}{8} \, x^{16} + \frac {154}{5} \, x^{15} + 33 \, x^{14} + \frac {330}{13} \, x^{13} + \frac {55}{4} \, x^{12} + 5 \, x^{11} + \frac {11}{10} \, x^{10} + \frac {1}{9} \, x^{9} \]
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Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{20} \, x^{20} + \frac {11}{19} \, x^{19} + \frac {55}{18} \, x^{18} + \frac {165}{17} \, x^{17} + \frac {165}{8} \, x^{16} + \frac {154}{5} \, x^{15} + 33 \, x^{14} + \frac {330}{13} \, x^{13} + \frac {55}{4} \, x^{12} + 5 \, x^{11} + \frac {11}{10} \, x^{10} + \frac {1}{9} \, x^{9} \]
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Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{20}}{20}+\frac {11\,x^{19}}{19}+\frac {55\,x^{18}}{18}+\frac {165\,x^{17}}{17}+\frac {165\,x^{16}}{8}+\frac {154\,x^{15}}{5}+33\,x^{14}+\frac {330\,x^{13}}{13}+\frac {55\,x^{12}}{4}+5\,x^{11}+\frac {11\,x^{10}}{10}+\frac {x^9}{9} \]
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