Integrand size = 27, antiderivative size = 79 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=\frac {65}{32} (3+2 x)^{5/2}-\frac {1065}{224} (3+2 x)^{7/2}+\frac {217}{48} (3+2 x)^{9/2}-\frac {359}{176} (3+2 x)^{11/2}+\frac {165}{416} (3+2 x)^{13/2}-\frac {3}{160} (3+2 x)^{15/2} \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {785} \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {3}{160} (2 x+3)^{15/2}+\frac {165}{416} (2 x+3)^{13/2}-\frac {359}{176} (2 x+3)^{11/2}+\frac {217}{48} (2 x+3)^{9/2}-\frac {1065}{224} (2 x+3)^{7/2}+\frac {65}{32} (2 x+3)^{5/2} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {325}{32} (3+2 x)^{3/2}-\frac {1065}{32} (3+2 x)^{5/2}+\frac {651}{16} (3+2 x)^{7/2}-\frac {359}{16} (3+2 x)^{9/2}+\frac {165}{32} (3+2 x)^{11/2}-\frac {9}{32} (3+2 x)^{13/2}\right ) \, dx \\ & = \frac {65}{32} (3+2 x)^{5/2}-\frac {1065}{224} (3+2 x)^{7/2}+\frac {217}{48} (3+2 x)^{9/2}-\frac {359}{176} (3+2 x)^{11/2}+\frac {165}{416} (3+2 x)^{13/2}-\frac {3}{160} (3+2 x)^{15/2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {(3+2 x)^{5/2} \left (-14304-76260 x-151270 x^2-124005 x^3-27720 x^4+9009 x^5\right )}{15015} \]
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Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (9009 x^{5}-27720 x^{4}-124005 x^{3}-151270 x^{2}-76260 x -14304\right ) \left (3+2 x \right )^{\frac {5}{2}}}{15015}\) | \(35\) |
pseudoelliptic | \(-\frac {12 \left (x^{5}-\frac {40}{13} x^{4}-\frac {5905}{429} x^{3}-\frac {21610}{1287} x^{2}-\frac {25420}{3003} x -\frac {4768}{3003}\right ) \sqrt {3+2 x}\, \left (x +\frac {3}{2}\right )^{2}}{5}\) | \(38\) |
trager | \(\left (-\frac {12}{5} x^{7}+\frac {12}{65} x^{6}+\frac {35599}{715} x^{5}+\frac {66932}{429} x^{4}+\frac {215755}{1001} x^{3}+\frac {777922}{5005} x^{2}+\frac {285996}{5005} x +\frac {42912}{5005}\right ) \sqrt {3+2 x}\) | \(44\) |
risch | \(-\frac {\left (36036 x^{7}-2772 x^{6}-747579 x^{5}-2342620 x^{4}-3236325 x^{3}-2333766 x^{2}-857988 x -128736\right ) \sqrt {3+2 x}}{15015}\) | \(45\) |
derivativedivides | \(\frac {65 \left (3+2 x \right )^{\frac {5}{2}}}{32}-\frac {1065 \left (3+2 x \right )^{\frac {7}{2}}}{224}+\frac {217 \left (3+2 x \right )^{\frac {9}{2}}}{48}-\frac {359 \left (3+2 x \right )^{\frac {11}{2}}}{176}+\frac {165 \left (3+2 x \right )^{\frac {13}{2}}}{416}-\frac {3 \left (3+2 x \right )^{\frac {15}{2}}}{160}\) | \(56\) |
default | \(\frac {65 \left (3+2 x \right )^{\frac {5}{2}}}{32}-\frac {1065 \left (3+2 x \right )^{\frac {7}{2}}}{224}+\frac {217 \left (3+2 x \right )^{\frac {9}{2}}}{48}-\frac {359 \left (3+2 x \right )^{\frac {11}{2}}}{176}+\frac {165 \left (3+2 x \right )^{\frac {13}{2}}}{416}-\frac {3 \left (3+2 x \right )^{\frac {15}{2}}}{160}\) | \(56\) |
meijerg | \(\frac {40095 \sqrt {3}\, \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (\frac {1120}{81} x^{4}+\frac {800}{27} x^{3}+\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{945}\right )}{32 \sqrt {\pi }}+\frac {82377 \sqrt {3}\, \left (\frac {128 \sqrt {\pi }}{3465}-\frac {\sqrt {\pi }\, \left (-\frac {8960}{81} x^{5}-\frac {17920}{81} x^{4}-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{3465}\right )}{64 \sqrt {\pi }}+\frac {32805 \sqrt {3}\, \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (\frac {49280}{243} x^{6}+\frac {31360}{81} x^{5}+\frac {1120}{81} x^{4}-\frac {640}{27} x^{3}+\frac {128}{3} x^{2}-\frac {256}{3} x +256\right ) \sqrt {1+\frac {2 x}{3}}}{45045}\right )}{128 \sqrt {\pi }}+\frac {486 \sqrt {3}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {2 \sqrt {\pi }\, \left (-\frac {160}{27} x^{3}-\frac {128}{9} x^{2}-\frac {8}{3} x +8\right ) \sqrt {1+\frac {2 x}{3}}}{105}\right )}{\sqrt {\pi }}+\frac {135 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {8}{9} x^{2}+\frac {8}{3} x +2\right ) \sqrt {1+\frac {2 x}{3}}}{15}\right )}{2 \sqrt {\pi }}-\frac {59049 \sqrt {3}\, \left (\frac {2048 \sqrt {\pi }}{135135}-\frac {2 \sqrt {\pi }\, \left (-\frac {512512}{729} x^{7}-\frac {315392}{243} x^{6}-\frac {896}{27} x^{5}+\frac {4480}{81} x^{4}-\frac {2560}{27} x^{3}+\frac {512}{3} x^{2}-\frac {1024}{3} x +1024\right ) \sqrt {1+\frac {2 x}{3}}}{135135}\right )}{256 \sqrt {\pi }}\) | \(293\) |
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Time = 0.39 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {1}{15015} \, {\left (36036 \, x^{7} - 2772 \, x^{6} - 747579 \, x^{5} - 2342620 \, x^{4} - 3236325 \, x^{3} - 2333766 \, x^{2} - 857988 \, x - 128736\right )} \sqrt {2 \, x + 3} \]
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Time = 0.99 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=- \frac {3 \left (2 x + 3\right )^{\frac {15}{2}}}{160} + \frac {165 \left (2 x + 3\right )^{\frac {13}{2}}}{416} - \frac {359 \left (2 x + 3\right )^{\frac {11}{2}}}{176} + \frac {217 \left (2 x + 3\right )^{\frac {9}{2}}}{48} - \frac {1065 \left (2 x + 3\right )^{\frac {7}{2}}}{224} + \frac {65 \left (2 x + 3\right )^{\frac {5}{2}}}{32} \]
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Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {3}{160} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {165}{416} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {359}{176} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {217}{48} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {1065}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {65}{32} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {3}{160} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {165}{416} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {359}{176} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {217}{48} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {1065}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {65}{32} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2 \, dx=\frac {65\,{\left (2\,x+3\right )}^{5/2}}{32}-\frac {1065\,{\left (2\,x+3\right )}^{7/2}}{224}+\frac {217\,{\left (2\,x+3\right )}^{9/2}}{48}-\frac {359\,{\left (2\,x+3\right )}^{11/2}}{176}+\frac {165\,{\left (2\,x+3\right )}^{13/2}}{416}-\frac {3\,{\left (2\,x+3\right )}^{15/2}}{160} \]
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