Integrand size = 19, antiderivative size = 27 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b (1+12 m)} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {267} \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b (1+12 m)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b+156 b m} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(25)=50\).
Time = 45.19 (sec) , antiderivative size = 288, normalized size of antiderivative = 10.67
method | result | size |
parallelrisch | \(\frac {b^{12} x^{12+144 m} x^{12 m} x +13 a \,b^{11} x^{11+132 m} x^{12 m} x +78 a^{2} b^{10} x^{10+120 m} x^{12 m} x +286 a^{3} b^{9} x^{9+108 m} x^{12 m} x +715 a^{4} b^{8} x^{8+96 m} x^{12 m} x +1287 a^{5} b^{7} x^{7+84 m} x^{12 m} x +1716 a^{6} b^{6} x^{6+72 m} x^{12 m} x +1716 a^{7} b^{5} x^{5+60 m} x^{12 m} x +1287 a^{8} b^{4} x^{4+48 m} x^{12 m} x +715 a^{9} b^{3} x^{3+36 m} x^{12 m} x +286 a^{10} b^{2} x^{2+24 m} x^{12 m} x +78 b \,a^{11} x^{1+12 m} x^{12 m} x +13 a^{12} x^{12 m} x}{13+156 m}\) | \(288\) |
risch | \(\frac {b^{12} x^{13} x^{156 m}}{13+156 m}+\frac {a \,b^{11} x^{12} x^{144 m}}{1+12 m}+\frac {6 a^{2} b^{10} x^{11} x^{132 m}}{1+12 m}+\frac {22 a^{3} b^{9} x^{10} x^{120 m}}{1+12 m}+\frac {55 a^{4} b^{8} x^{9} x^{108 m}}{1+12 m}+\frac {99 a^{5} b^{7} x^{8} x^{96 m}}{1+12 m}+\frac {132 a^{6} b^{6} x^{7} x^{84 m}}{1+12 m}+\frac {132 a^{7} b^{5} x^{6} x^{72 m}}{1+12 m}+\frac {99 a^{8} b^{4} x^{5} x^{60 m}}{1+12 m}+\frac {55 a^{9} b^{3} x^{4} x^{48 m}}{1+12 m}+\frac {22 a^{10} b^{2} x^{3} x^{36 m}}{1+12 m}+\frac {6 b \,a^{11} x^{2} x^{24 m}}{1+12 m}+\frac {a^{12} x \,x^{12 m}}{1+12 m}\) | \(311\) |
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 7.19 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {b^{12} x^{156 \, m + 13} + 13 \, a b^{11} x^{144 \, m + 12} + 78 \, a^{2} b^{10} x^{132 \, m + 11} + 286 \, a^{3} b^{9} x^{120 \, m + 10} + 715 \, a^{4} b^{8} x^{108 \, m + 9} + 1287 \, a^{5} b^{7} x^{96 \, m + 8} + 1716 \, a^{6} b^{6} x^{84 \, m + 7} + 1716 \, a^{7} b^{5} x^{72 \, m + 6} + 1287 \, a^{8} b^{4} x^{60 \, m + 5} + 715 \, a^{9} b^{3} x^{48 \, m + 4} + 286 \, a^{10} b^{2} x^{36 \, m + 3} + 78 \, a^{11} b x^{24 \, m + 2} + 13 \, a^{12} x^{12 \, m + 1}}{13 \, {\left (12 \, m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (19) = 38\).
Time = 6.22 (sec) , antiderivative size = 345, normalized size of antiderivative = 12.78 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\begin {cases} \frac {13 a^{12} x x^{12 m}}{156 m + 13} + \frac {78 a^{11} b x x^{12 m} x^{12 m + 1}}{156 m + 13} + \frac {286 a^{10} b^{2} x x^{12 m} x^{24 m + 2}}{156 m + 13} + \frac {715 a^{9} b^{3} x x^{12 m} x^{36 m + 3}}{156 m + 13} + \frac {1287 a^{8} b^{4} x x^{12 m} x^{48 m + 4}}{156 m + 13} + \frac {1716 a^{7} b^{5} x x^{12 m} x^{60 m + 5}}{156 m + 13} + \frac {1716 a^{6} b^{6} x x^{12 m} x^{72 m + 6}}{156 m + 13} + \frac {1287 a^{5} b^{7} x x^{12 m} x^{84 m + 7}}{156 m + 13} + \frac {715 a^{4} b^{8} x x^{12 m} x^{96 m + 8}}{156 m + 13} + \frac {286 a^{3} b^{9} x x^{12 m} x^{108 m + 9}}{156 m + 13} + \frac {78 a^{2} b^{10} x x^{12 m} x^{120 m + 10}}{156 m + 13} + \frac {13 a b^{11} x x^{12 m} x^{132 m + 11}}{156 m + 13} + \frac {b^{12} x x^{12 m} x^{144 m + 12}}{156 m + 13} & \text {for}\: m \neq - \frac {1}{12} \\\left (a + b\right )^{12} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {{\left (b x^{12 \, m + 1} + a\right )}^{13}}{13 \, b {\left (12 \, m + 1\right )}} \]
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Time = 0.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {{\left (b x^{12 \, m + 1} + a\right )}^{13}}{13 \, b {\left (12 \, m + 1\right )}} \]
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Time = 6.58 (sec) , antiderivative size = 285, normalized size of antiderivative = 10.56 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {b^{12}\,x^{156\,m}\,x^{13}}{156\,m+13}+\frac {a^{12}\,x\,x^{12\,m}}{12\,m+1}+\frac {6\,a^{11}\,b\,x^{24\,m}\,x^2}{12\,m+1}+\frac {a\,b^{11}\,x^{144\,m}\,x^{12}}{12\,m+1}+\frac {22\,a^{10}\,b^2\,x^{36\,m}\,x^3}{12\,m+1}+\frac {55\,a^9\,b^3\,x^{48\,m}\,x^4}{12\,m+1}+\frac {99\,a^8\,b^4\,x^{60\,m}\,x^5}{12\,m+1}+\frac {132\,a^7\,b^5\,x^{72\,m}\,x^6}{12\,m+1}+\frac {132\,a^6\,b^6\,x^{84\,m}\,x^7}{12\,m+1}+\frac {99\,a^5\,b^7\,x^{96\,m}\,x^8}{12\,m+1}+\frac {55\,a^4\,b^8\,x^{108\,m}\,x^9}{12\,m+1}+\frac {22\,a^3\,b^9\,x^{120\,m}\,x^{10}}{12\,m+1}+\frac {6\,a^2\,b^{10}\,x^{132\,m}\,x^{11}}{12\,m+1} \]
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