Integrand size = 17, antiderivative size = 40 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {121}{28} (1-2 x)^{7/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {25}{44} (1-2 x)^{11/2} \]
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Time = 0.01 (sec) , antiderivative size = 40, 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 = {45} \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {25}{44} (1-2 x)^{11/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {121}{28} (1-2 x)^{7/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {121}{4} (1-2 x)^{5/2}-\frac {55}{2} (1-2 x)^{7/2}+\frac {25}{4} (1-2 x)^{9/2}\right ) \, dx \\ & = -\frac {121}{28} (1-2 x)^{7/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {25}{44} (1-2 x)^{11/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {1}{693} (1-2 x)^{7/2} \left (1271+2660 x+1575 x^2\right ) \]
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Time = 1.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (1575 x^{2}+2660 x +1271\right )}{693}\) | \(20\) |
pseudoelliptic | \(\frac {\sqrt {1-2 x}\, \left (-1+2 x \right )^{3} \left (1575 x^{2}+2660 x +1271\right )}{693}\) | \(27\) |
derivativedivides | \(-\frac {121 \left (1-2 x \right )^{\frac {7}{2}}}{28}+\frac {55 \left (1-2 x \right )^{\frac {9}{2}}}{18}-\frac {25 \left (1-2 x \right )^{\frac {11}{2}}}{44}\) | \(29\) |
default | \(-\frac {121 \left (1-2 x \right )^{\frac {7}{2}}}{28}+\frac {55 \left (1-2 x \right )^{\frac {9}{2}}}{18}-\frac {25 \left (1-2 x \right )^{\frac {11}{2}}}{44}\) | \(29\) |
trager | \(\left (\frac {200}{11} x^{5}+\frac {340}{99} x^{4}-\frac {12302}{693} x^{3}-\frac {289}{231} x^{2}+\frac {4966}{693} x -\frac {1271}{693}\right ) \sqrt {1-2 x}\) | \(34\) |
risch | \(-\frac {\left (12600 x^{5}+2380 x^{4}-12302 x^{3}-867 x^{2}+4966 x -1271\right ) \left (-1+2 x \right )}{693 \sqrt {1-2 x}}\) | \(40\) |
meijerg | \(\frac {\frac {9 \sqrt {\pi }}{7}-\frac {9 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{14}}{\sqrt {\pi }}-\frac {225 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{16 \sqrt {\pi }}+\frac {\frac {50 \sqrt {\pi }}{693}-\frac {25 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{5544}}{\sqrt {\pi }}\) | \(131\) |
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=\frac {1}{693} \, {\left (12600 \, x^{5} + 2380 \, x^{4} - 12302 \, x^{3} - 867 \, x^{2} + 4966 \, x - 1271\right )} \sqrt {-2 \, x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (34) = 68\).
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.12 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=\frac {200 x^{5} \sqrt {1 - 2 x}}{11} + \frac {340 x^{4} \sqrt {1 - 2 x}}{99} - \frac {12302 x^{3} \sqrt {1 - 2 x}}{693} - \frac {289 x^{2} \sqrt {1 - 2 x}}{231} + \frac {4966 x \sqrt {1 - 2 x}}{693} - \frac {1271 \sqrt {1 - 2 x}}{693} \]
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Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {25}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {55}{18} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {121}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \]
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Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.22 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=\frac {25}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {55}{18} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {121}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \]
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Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (1-2 x)^{5/2} (3+5 x)^2 \, dx=-\frac {{\left (1-2\,x\right )}^{7/2}\,\left (16940\,x+1575\,{\left (2\,x-1\right )}^2+3509\right )}{2772} \]
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