Integrand size = 17, antiderivative size = 45 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=x+4 x^2+8 x^3+8 x^4+\frac {24 x^5}{5}+\frac {16 x^6}{3}+\frac {32 x^7}{7}+\frac {16 x^9}{9} \]
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Time = 0.01 (sec) , antiderivative size = 45, 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} \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=\frac {16 x^9}{9}+\frac {32 x^7}{7}+\frac {16 x^6}{3}+\frac {24 x^5}{5}+8 x^4+8 x^3+4 x^2+x \]
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Rule 2086
Rubi steps \begin{align*} \text {integral}& = \int \left (1+8 x+24 x^2+32 x^3+24 x^4+32 x^5+32 x^6+16 x^8\right ) \, dx \\ & = x+4 x^2+8 x^3+8 x^4+\frac {24 x^5}{5}+\frac {16 x^6}{3}+\frac {32 x^7}{7}+\frac {16 x^9}{9} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=x+4 x^2+8 x^3+8 x^4+\frac {24 x^5}{5}+\frac {16 x^6}{3}+\frac {32 x^7}{7}+\frac {16 x^9}{9} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(x +4 x^{2}+8 x^{3}+8 x^{4}+\frac {24}{5} x^{5}+\frac {16}{3} x^{6}+\frac {32}{7} x^{7}+\frac {16}{9} x^{9}\) | \(38\) |
default | \(x +4 x^{2}+8 x^{3}+8 x^{4}+\frac {24}{5} x^{5}+\frac {16}{3} x^{6}+\frac {32}{7} x^{7}+\frac {16}{9} x^{9}\) | \(38\) |
norman | \(x +4 x^{2}+8 x^{3}+8 x^{4}+\frac {24}{5} x^{5}+\frac {16}{3} x^{6}+\frac {32}{7} x^{7}+\frac {16}{9} x^{9}\) | \(38\) |
risch | \(x +4 x^{2}+8 x^{3}+8 x^{4}+\frac {24}{5} x^{5}+\frac {16}{3} x^{6}+\frac {32}{7} x^{7}+\frac {16}{9} x^{9}\) | \(38\) |
parallelrisch | \(x +4 x^{2}+8 x^{3}+8 x^{4}+\frac {24}{5} x^{5}+\frac {16}{3} x^{6}+\frac {32}{7} x^{7}+\frac {16}{9} x^{9}\) | \(38\) |
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=\frac {16}{9} \, x^{9} + \frac {32}{7} \, x^{7} + \frac {16}{3} \, x^{6} + \frac {24}{5} \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, x^{2} + x \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=\frac {16 x^{9}}{9} + \frac {32 x^{7}}{7} + \frac {16 x^{6}}{3} + \frac {24 x^{5}}{5} + 8 x^{4} + 8 x^{3} + 4 x^{2} + x \]
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none
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=\frac {16}{9} \, x^{9} + \frac {32}{7} \, x^{7} + \frac {16}{3} \, x^{6} + \frac {24}{5} \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, x^{2} + x \]
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Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=\frac {16}{9} \, x^{9} + \frac {32}{7} \, x^{7} + \frac {16}{3} \, x^{6} + \frac {24}{5} \, x^{5} + 8 \, x^{4} + 8 \, x^{3} + 4 \, x^{2} + x \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \left (1+4 x+4 x^2+4 x^4\right )^2 \, dx=\frac {16\,x^9}{9}+\frac {32\,x^7}{7}+\frac {16\,x^6}{3}+\frac {24\,x^5}{5}+8\,x^4+8\,x^3+4\,x^2+x \]
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