Integrand size = 23, antiderivative size = 27 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b (1+12 m)} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1598, 267} \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \]
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Rule 267
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int x^{12+12 (-1+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)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b+156 b m} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(337\) vs. \(2(25)=50\).
Time = 53.75 (sec) , antiderivative size = 338, normalized size of antiderivative = 12.52
method | result | size |
parallelrisch | \(\frac {13 a^{12} x^{-12+12 m} x^{13}+78 b \,a^{11} x^{-12+12 m} x^{2+12 m} x^{12}+286 a^{10} b^{2} x^{-12+12 m} x^{4+24 m} x^{11}+715 a^{9} b^{3} x^{-12+12 m} x^{6+36 m} x^{10}+1287 a^{8} b^{4} x^{-12+12 m} x^{8+48 m} x^{9}+1716 a^{7} b^{5} x^{-12+12 m} x^{10+60 m} x^{8}+1716 a^{6} b^{6} x^{-12+12 m} x^{12+72 m} x^{7}+1287 a^{5} b^{7} x^{-12+12 m} x^{14+84 m} x^{6}+715 a^{4} b^{8} x^{-12+12 m} x^{16+96 m} x^{5}+286 a^{3} b^{9} x^{-12+12 m} x^{18+108 m} x^{4}+78 a^{2} b^{10} x^{-12+12 m} x^{20+120 m} x^{3}+13 a \,b^{11} x^{-12+12 m} x^{22+132 m} x^{2}+b^{12} x^{-12+12 m} x^{24+144 m} x}{13+156 m}\) | \(338\) |
risch | \(\frac {b^{12} x^{26+156 m}}{13 \left (1+12 m \right ) x^{13}}+\frac {a \,b^{11} x^{24+144 m}}{\left (1+12 m \right ) x^{12}}+\frac {6 a^{2} b^{10} x^{22+132 m}}{\left (1+12 m \right ) x^{11}}+\frac {22 a^{3} b^{9} x^{20+120 m}}{\left (1+12 m \right ) x^{10}}+\frac {55 a^{4} b^{8} x^{18+108 m}}{\left (1+12 m \right ) x^{9}}+\frac {99 a^{5} b^{7} x^{16+96 m}}{\left (1+12 m \right ) x^{8}}+\frac {132 a^{6} b^{6} x^{14+84 m}}{\left (1+12 m \right ) x^{7}}+\frac {132 a^{7} b^{5} x^{12+72 m}}{\left (1+12 m \right ) x^{6}}+\frac {99 a^{8} b^{4} x^{10+60 m}}{\left (1+12 m \right ) x^{5}}+\frac {55 a^{9} b^{3} x^{8+48 m}}{\left (1+12 m \right ) x^{4}}+\frac {22 a^{10} b^{2} x^{6+36 m}}{\left (1+12 m \right ) x^{3}}+\frac {6 b \,a^{11} x^{4+24 m}}{\left (1+12 m \right ) x^{2}}+\frac {a^{12} x^{2+12 m}}{\left (1+12 m \right ) x}\) | \(339\) |
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 231, normalized size of antiderivative = 8.56 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {13 \, a^{12} x^{12} x^{12 \, m + 2} + 78 \, a^{11} b x^{11} x^{24 \, m + 4} + 286 \, a^{10} b^{2} x^{10} x^{36 \, m + 6} + 715 \, a^{9} b^{3} x^{9} x^{48 \, m + 8} + 1287 \, a^{8} b^{4} x^{8} x^{60 \, m + 10} + 1716 \, a^{7} b^{5} x^{7} x^{72 \, m + 12} + 1716 \, a^{6} b^{6} x^{6} x^{84 \, m + 14} + 1287 \, a^{5} b^{7} x^{5} x^{96 \, m + 16} + 715 \, a^{4} b^{8} x^{4} x^{108 \, m + 18} + 286 \, a^{3} b^{9} x^{3} x^{120 \, m + 20} + 78 \, a^{2} b^{10} x^{2} x^{132 \, m + 22} + 13 \, a b^{11} x x^{144 \, m + 24} + b^{12} x^{156 \, m + 26}}{13 \, {\left (12 \, m + 1\right )} x^{13}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (19) = 38\).
Time = 6.30 (sec) , antiderivative size = 520, normalized size of antiderivative = 19.26 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\begin {cases} \frac {13 a^{12} x^{13} x^{12 m - 12}}{156 m + 13} + \frac {78 a^{11} b x^{12} x^{12 m - 12} x^{12 m + 2}}{156 m + 13} + \frac {286 a^{10} b^{2} x^{11} x^{12 m - 12} x^{24 m + 4}}{156 m + 13} + \frac {715 a^{9} b^{3} x^{10} x^{12 m - 12} x^{36 m + 6}}{156 m + 13} + \frac {1287 a^{8} b^{4} x^{9} x^{12 m - 12} x^{48 m + 8}}{156 m + 13} + \frac {1716 a^{7} b^{5} x^{8} x^{12 m - 12} x^{60 m + 10}}{156 m + 13} + \frac {1716 a^{6} b^{6} x^{7} x^{12 m - 12} x^{72 m + 12}}{156 m + 13} + \frac {1287 a^{5} b^{7} x^{6} x^{12 m - 12} x^{84 m + 14}}{156 m + 13} + \frac {715 a^{4} b^{8} x^{5} x^{12 m - 12} x^{96 m + 16}}{156 m + 13} + \frac {286 a^{3} b^{9} x^{4} x^{12 m - 12} x^{108 m + 18}}{156 m + 13} + \frac {78 a^{2} b^{10} x^{3} x^{12 m - 12} x^{120 m + 20}}{156 m + 13} + \frac {13 a b^{11} x^{2} x^{12 m - 12} x^{132 m + 22}}{156 m + 13} + \frac {b^{12} x x^{12 m - 12} x^{144 m + 24}}{156 m + 13} & \text {for}\: m \neq - \frac {1}{12} \\a^{12} \log {\left (x \right )} + 12 a^{11} b \log {\left (x \right )} + 66 a^{10} b^{2} \log {\left (x \right )} + 220 a^{9} b^{3} \log {\left (x \right )} + 495 a^{8} b^{4} \log {\left (x \right )} + 792 a^{7} b^{5} \log {\left (x \right )} + 924 a^{6} b^{6} \log {\left (x \right )} + 792 a^{5} b^{7} \log {\left (x \right )} + 495 a^{4} b^{8} \log {\left (x \right )} + 220 a^{3} b^{9} \log {\left (x \right )} + 66 a^{2} b^{10} \log {\left (x \right )} + 12 a b^{11} \log {\left (x \right )} + b^{12} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 275, normalized size of antiderivative = 10.19 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {b^{12} x^{156 \, m + 13}}{13 \, {\left (12 \, m + 1\right )}} + \frac {a b^{11} x^{144 \, m + 12}}{12 \, m + 1} + \frac {6 \, a^{2} b^{10} x^{132 \, m + 11}}{12 \, m + 1} + \frac {22 \, a^{3} b^{9} x^{120 \, m + 10}}{12 \, m + 1} + \frac {55 \, a^{4} b^{8} x^{108 \, m + 9}}{12 \, m + 1} + \frac {99 \, a^{5} b^{7} x^{96 \, m + 8}}{12 \, m + 1} + \frac {132 \, a^{6} b^{6} x^{84 \, m + 7}}{12 \, m + 1} + \frac {132 \, a^{7} b^{5} x^{72 \, m + 6}}{12 \, m + 1} + \frac {99 \, a^{8} b^{4} x^{60 \, m + 5}}{12 \, m + 1} + \frac {55 \, a^{9} b^{3} x^{48 \, m + 4}}{12 \, m + 1} + \frac {22 \, a^{10} b^{2} x^{36 \, m + 3}}{12 \, m + 1} + \frac {6 \, a^{11} b x^{24 \, m + 2}}{12 \, m + 1} + \frac {a^{12} x^{12 \, m + 1}}{12 \, m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (25) = 50\).
Time = 0.40 (sec) , antiderivative size = 285, normalized size of antiderivative = 10.56 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {13 \, a^{12} x^{12} e^{\left (12 \, m \log \left (x\right ) + 2 \, \log \left (x\right )\right )} + 78 \, a^{11} b x^{11} e^{\left (24 \, m \log \left (x\right ) + 4 \, \log \left (x\right )\right )} + 286 \, a^{10} b^{2} x^{10} e^{\left (36 \, m \log \left (x\right ) + 6 \, \log \left (x\right )\right )} + 715 \, a^{9} b^{3} x^{9} e^{\left (48 \, m \log \left (x\right ) + 8 \, \log \left (x\right )\right )} + 1287 \, a^{8} b^{4} x^{8} e^{\left (60 \, m \log \left (x\right ) + 10 \, \log \left (x\right )\right )} + 1716 \, a^{7} b^{5} x^{7} e^{\left (72 \, m \log \left (x\right ) + 12 \, \log \left (x\right )\right )} + 1716 \, a^{6} b^{6} x^{6} e^{\left (84 \, m \log \left (x\right ) + 14 \, \log \left (x\right )\right )} + 1287 \, a^{5} b^{7} x^{5} e^{\left (96 \, m \log \left (x\right ) + 16 \, \log \left (x\right )\right )} + 715 \, a^{4} b^{8} x^{4} e^{\left (108 \, m \log \left (x\right ) + 18 \, \log \left (x\right )\right )} + 286 \, a^{3} b^{9} x^{3} e^{\left (120 \, m \log \left (x\right ) + 20 \, \log \left (x\right )\right )} + 78 \, a^{2} b^{10} x^{2} e^{\left (132 \, m \log \left (x\right ) + 22 \, \log \left (x\right )\right )} + 13 \, a b^{11} x e^{\left (144 \, m \log \left (x\right ) + 24 \, \log \left (x\right )\right )} + b^{12} e^{\left (156 \, m \log \left (x\right ) + 26 \, \log \left (x\right )\right )}}{13 \, {\left (12 \, m x^{13} + x^{13}\right )}} \]
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Time = 9.85 (sec) , antiderivative size = 287, normalized size of antiderivative = 10.63 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {b^{12}\,x^{156\,m}\,x^{13}}{156\,m+13}+\frac {13\,a^{12}\,x\,x^{12\,m}}{156\,m+13}+\frac {78\,a^{11}\,b\,x^{24\,m}\,x^2}{156\,m+13}+\frac {13\,a\,b^{11}\,x^{144\,m}\,x^{12}}{156\,m+13}+\frac {286\,a^{10}\,b^2\,x^{36\,m}\,x^3}{156\,m+13}+\frac {715\,a^9\,b^3\,x^{48\,m}\,x^4}{156\,m+13}+\frac {1287\,a^8\,b^4\,x^{60\,m}\,x^5}{156\,m+13}+\frac {1716\,a^7\,b^5\,x^{72\,m}\,x^6}{156\,m+13}+\frac {1716\,a^6\,b^6\,x^{84\,m}\,x^7}{156\,m+13}+\frac {1287\,a^5\,b^7\,x^{96\,m}\,x^8}{156\,m+13}+\frac {715\,a^4\,b^8\,x^{108\,m}\,x^9}{156\,m+13}+\frac {286\,a^3\,b^9\,x^{120\,m}\,x^{10}}{156\,m+13}+\frac {78\,a^2\,b^{10}\,x^{132\,m}\,x^{11}}{156\,m+13} \]
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