Integrand size = 20, antiderivative size = 40 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=-\frac {11}{4} (1-2 x)^{7/2}+\frac {17}{9} (1-2 x)^{9/2}-\frac {15}{44} (1-2 x)^{11/2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=-\frac {15}{44} (1-2 x)^{11/2}+\frac {17}{9} (1-2 x)^{9/2}-\frac {11}{4} (1-2 x)^{7/2} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {77}{4} (1-2 x)^{5/2}-17 (1-2 x)^{7/2}+\frac {15}{4} (1-2 x)^{9/2}\right ) \, dx \\ & = -\frac {11}{4} (1-2 x)^{7/2}+\frac {17}{9} (1-2 x)^{9/2}-\frac {15}{44} (1-2 x)^{11/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=-\frac {1}{99} (1-2 x)^{7/2} \left (119+239 x+135 x^2\right ) \]
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Time = 0.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (135 x^{2}+239 x +119\right )}{99}\) | \(20\) |
pseudoelliptic | \(\frac {\left (135 x^{2}+239 x +119\right ) \sqrt {1-2 x}\, \left (-1+2 x \right )^{3}}{99}\) | \(27\) |
derivativedivides | \(-\frac {11 \left (1-2 x \right )^{\frac {7}{2}}}{4}+\frac {17 \left (1-2 x \right )^{\frac {9}{2}}}{9}-\frac {15 \left (1-2 x \right )^{\frac {11}{2}}}{44}\) | \(29\) |
default | \(-\frac {11 \left (1-2 x \right )^{\frac {7}{2}}}{4}+\frac {17 \left (1-2 x \right )^{\frac {9}{2}}}{9}-\frac {15 \left (1-2 x \right )^{\frac {11}{2}}}{44}\) | \(29\) |
trager | \(\left (\frac {120}{11} x^{5}+\frac {292}{99} x^{4}-\frac {1106}{99} x^{3}-\frac {43}{33} x^{2}+\frac {475}{99} x -\frac {119}{99}\right ) \sqrt {1-2 x}\) | \(34\) |
risch | \(-\frac {\left (1080 x^{5}+292 x^{4}-1106 x^{3}-129 x^{2}+475 x -119\right ) \left (-1+2 x \right )}{99 \sqrt {1-2 x}}\) | \(40\) |
meijerg | \(\frac {\frac {6 \sqrt {\pi }}{7}-\frac {3 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {285 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}+\frac {\frac {10 \sqrt {\pi }}{231}-\frac {5 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{1848}}{\sqrt {\pi }}\) | \(131\) |
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=\frac {1}{99} \, {\left (1080 \, x^{5} + 292 \, x^{4} - 1106 \, x^{3} - 129 \, x^{2} + 475 \, x - 119\right )} \sqrt {-2 \, x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (34) = 68\).
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.12 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=\frac {120 x^{5} \sqrt {1 - 2 x}}{11} + \frac {292 x^{4} \sqrt {1 - 2 x}}{99} - \frac {1106 x^{3} \sqrt {1 - 2 x}}{99} - \frac {43 x^{2} \sqrt {1 - 2 x}}{33} + \frac {475 x \sqrt {1 - 2 x}}{99} - \frac {119 \sqrt {1 - 2 x}}{99} \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=-\frac {15}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {17}{9} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {11}{4} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.22 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=\frac {15}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {17}{9} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {11}{4} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \]
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Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx=-\frac {{\left (1-2\,x\right )}^{7/2}\,\left (1496\,x+135\,{\left (2\,x-1\right )}^2+341\right )}{396} \]
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