Integrand size = 17, antiderivative size = 97 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=x+8 x^2+\frac {112 x^3}{3}+112 x^4+\frac {1136 x^5}{5}+\frac {992 x^6}{3}+\frac {2752 x^7}{7}+448 x^8+\frac {4192 x^9}{9}+384 x^{10}+\frac {3328 x^{11}}{11}+256 x^{12}+\frac {1792 x^{13}}{13}+\frac {512 x^{14}}{7}+\frac {1024 x^{15}}{15}+\frac {256 x^{17}}{17} \]
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Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2086} \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256 x^{17}}{17}+\frac {1024 x^{15}}{15}+\frac {512 x^{14}}{7}+\frac {1792 x^{13}}{13}+256 x^{12}+\frac {3328 x^{11}}{11}+384 x^{10}+\frac {4192 x^9}{9}+448 x^8+\frac {2752 x^7}{7}+\frac {992 x^6}{3}+\frac {1136 x^5}{5}+112 x^4+\frac {112 x^3}{3}+8 x^2+x \]
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Rule 2086
Rubi steps \begin{align*} \text {integral}& = \int \left (1+16 x+112 x^2+448 x^3+1136 x^4+1984 x^5+2752 x^6+3584 x^7+4192 x^8+3840 x^9+3328 x^{10}+3072 x^{11}+1792 x^{12}+1024 x^{13}+1024 x^{14}+256 x^{16}\right ) \, 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} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=x+8 x^2+\frac {112 x^3}{3}+112 x^4+\frac {1136 x^5}{5}+\frac {992 x^6}{3}+\frac {2752 x^7}{7}+448 x^8+\frac {4192 x^9}{9}+384 x^{10}+\frac {3328 x^{11}}{11}+256 x^{12}+\frac {1792 x^{13}}{13}+\frac {512 x^{14}}{7}+\frac {1024 x^{15}}{15}+\frac {256 x^{17}}{17} \]
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Time = 0.03 (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 | \(x +8 x^{2}+\frac {112}{3} x^{3}+112 x^{4}+\frac {1136}{5} x^{5}+\frac {992}{3} x^{6}+\frac {2752}{7} x^{7}+448 x^{8}+\frac {4192}{9} x^{9}+384 x^{10}+\frac {3328}{11} x^{11}+256 x^{12}+\frac {1792}{13} x^{13}+\frac {512}{7} x^{14}+\frac {1024}{15} x^{15}+\frac {256}{17} x^{17}\) | \(78\) |
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256}{17} \, x^{17} + \frac {1024}{15} \, x^{15} + \frac {512}{7} \, x^{14} + \frac {1792}{13} \, x^{13} + 256 \, x^{12} + \frac {3328}{11} \, x^{11} + 384 \, x^{10} + \frac {4192}{9} \, x^{9} + 448 \, x^{8} + \frac {2752}{7} \, x^{7} + \frac {992}{3} \, x^{6} + \frac {1136}{5} \, x^{5} + 112 \, x^{4} + \frac {112}{3} \, x^{3} + 8 \, x^{2} + x \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256 x^{17}}{17} + \frac {1024 x^{15}}{15} + \frac {512 x^{14}}{7} + \frac {1792 x^{13}}{13} + 256 x^{12} + \frac {3328 x^{11}}{11} + 384 x^{10} + \frac {4192 x^{9}}{9} + 448 x^{8} + \frac {2752 x^{7}}{7} + \frac {992 x^{6}}{3} + \frac {1136 x^{5}}{5} + 112 x^{4} + \frac {112 x^{3}}{3} + 8 x^{2} + x \]
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Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256}{17} \, x^{17} + \frac {1024}{15} \, x^{15} + \frac {512}{7} \, x^{14} + \frac {1792}{13} \, x^{13} + 256 \, x^{12} + \frac {3328}{11} \, x^{11} + 384 \, x^{10} + \frac {4192}{9} \, x^{9} + 448 \, x^{8} + \frac {2752}{7} \, x^{7} + \frac {992}{3} \, x^{6} + \frac {1136}{5} \, x^{5} + 112 \, x^{4} + \frac {112}{3} \, x^{3} + 8 \, x^{2} + x \]
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Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256}{17} \, x^{17} + \frac {1024}{15} \, x^{15} + \frac {512}{7} \, x^{14} + \frac {1792}{13} \, x^{13} + 256 \, x^{12} + \frac {3328}{11} \, x^{11} + 384 \, x^{10} + \frac {4192}{9} \, x^{9} + 448 \, x^{8} + \frac {2752}{7} \, x^{7} + \frac {992}{3} \, x^{6} + \frac {1136}{5} \, x^{5} + 112 \, x^{4} + \frac {112}{3} \, x^{3} + 8 \, x^{2} + x \]
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Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^4 \, dx=\frac {256\,x^{17}}{17}+\frac {1024\,x^{15}}{15}+\frac {512\,x^{14}}{7}+\frac {1792\,x^{13}}{13}+256\,x^{12}+\frac {3328\,x^{11}}{11}+384\,x^{10}+\frac {4192\,x^9}{9}+448\,x^8+\frac {2752\,x^7}{7}+\frac {992\,x^6}{3}+\frac {1136\,x^5}{5}+112\,x^4+\frac {112\,x^3}{3}+8\,x^2+x \]
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