Integrand size = 26, antiderivative size = 194 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8192000000 \sqrt {10}} \]
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Time = 0.05 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {102, 152, 52, 56, 222} \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {355329559167 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8192000000 \sqrt {10}}-\frac {3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {735439 (5 x+3)^{3/2} (1-2 x)^{7/2}}{1280000}-\frac {9 (5 x+3)^{5/2} (13480 x+18399) (1-2 x)^{7/2}}{448000}-\frac {24269487 \sqrt {5 x+3} (1-2 x)^{7/2}}{20480000}+\frac {88988119 \sqrt {5 x+3} (1-2 x)^{5/2}}{204800000}+\frac {978869309 \sqrt {5 x+3} (1-2 x)^{3/2}}{819200000}+\frac {32302687197 \sqrt {5 x+3} \sqrt {1-2 x}}{8192000000} \]
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{80} \int \left (-323-\frac {1011 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx \\ & = -\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {735439 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{128000} \\ & = -\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {24269487 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{2560000} \\ & = -\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {266964357 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{40960000} \\ & = \frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {978869309 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{81920000} \\ & = \frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {32302687197 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1638400000} \\ & = \frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16384000000} \\ & = \frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8192000000 \sqrt {5}} \\ & = \frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8192000000 \sqrt {10}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-346249385613+2250324002625 x+8626817121940 x^2-517857728800 x^3-27529736688000 x^4-23895957760000 x^5+25359744000000 x^6+47139840000000 x^7+19353600000000 x^8\right )-2487306914169 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{573440000000 \sqrt {3+5 x}} \]
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Time = 1.14 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.63
method | result | size |
risch | \(-\frac {\left (3870720000000 x^{7}+7105536000000 x^{6}+808627200000 x^{5}-5264367872000 x^{4}-2347326614400 x^{3}+1304824422880 x^{2}+942468770660 x -115416461871\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{57344000000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {355329559167 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{163840000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(123\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (77414400000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+142110720000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+16172544000000 x^{5} \sqrt {-10 x^{2}-x +3}-105287357440000 x^{4} \sqrt {-10 x^{2}-x +3}-46946532288000 x^{3} \sqrt {-10 x^{2}-x +3}+26096488457600 x^{2} \sqrt {-10 x^{2}-x +3}+2487306914169 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+18849375413200 x \sqrt {-10 x^{2}-x +3}-2308329237420 \sqrt {-10 x^{2}-x +3}\right )}{1146880000000 \sqrt {-10 x^{2}-x +3}}\) | \(172\) |
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Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.47 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {1}{57344000000} \, {\left (3870720000000 \, x^{7} + 7105536000000 \, x^{6} + 808627200000 \, x^{5} - 5264367872000 \, x^{4} - 2347326614400 \, x^{3} + 1304824422880 \, x^{2} + 942468770660 \, x - 115416461871\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {355329559167}{163840000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.69 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {27}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {6183}{5600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {71331}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {6491477}{22400000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {8089829}{5120000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {8089829}{102400000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2936607927}{409600000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {355329559167}{163840000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {2936607927}{8192000000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (143) = 286\).
Time = 0.41 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.81 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {9}{573440000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 599\right )} {\left (5 \, x + 3\right )} + 175163\right )} {\left (5 \, x + 3\right )} - 4295993\right )} {\left (5 \, x + 3\right )} + 265620213\right )} {\left (5 \, x + 3\right )} - 2676516549\right )} {\left (5 \, x + 3\right )} + 35390483373\right )} {\left (5 \, x + 3\right )} - 164483997363\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 309625826895 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {99}{89600000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1029}{12800000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {457}{240000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {409}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {101}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {69}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {36}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
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Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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