Integrand size = 26, antiderivative size = 201 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=\frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1310720000 \sqrt {10}} \]
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Time = 0.05 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {102, 158, 152, 52, 56, 222} \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=\frac {1988505613369 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1310720000 \sqrt {10}}-\frac {3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac {1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac {3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac {135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac {1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {5 x+3}}{131072000}+\frac {180773237579 \sqrt {1-2 x} \sqrt {5 x+3}}{1310720000} \]
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 158
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {1}{80} \int \left (-452-\frac {1419 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx \\ & = -\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}+\frac {\int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \left (\frac {176225}{2}+\frac {548541 x}{4}\right ) \, dx}{5600} \\ & = -\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {135817609 \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx}{2560000} \\ & = -\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1493993699 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{8192000} \\ & = -\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {16433930689 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{32768000} \\ & = -\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {180773237579 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{262144000} \\ & = \frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2621440000} \\ & = \frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1310720000 \sqrt {5}} \\ & = \frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1310720000 \sqrt {10}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.49 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-17919913416273-41841631696875 x+690018024740 x^2+131786487855200 x^3+337325400912000 x^4+464971057920000 x^5+381250022400000 x^6+175094784000000 x^7+34836480000000 x^8\right )-41758617880749 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{275251200000 \sqrt {3+5 x}} \]
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Time = 1.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {\left (6967296000000 x^{7}+30838579200000 x^{6}+57746856960000 x^{5}+58346097408000 x^{4}+32457421737600 x^{3}+6882844528480 x^{2}-3991703112140 x -5973304472091\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{27525120000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1988505613369 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{26214400000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(123\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (139345920000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+616771584000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1154937139200000 x^{5} \sqrt {-10 x^{2}-x +3}+1166921948160000 x^{4} \sqrt {-10 x^{2}-x +3}+649148434752000 x^{3} \sqrt {-10 x^{2}-x +3}+137656890569600 x^{2} \sqrt {-10 x^{2}-x +3}+41758617880749 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-79834062242800 x \sqrt {-10 x^{2}-x +3}-119466089441820 \sqrt {-10 x^{2}-x +3}\right )}{550502400000 \sqrt {-10 x^{2}-x +3}}\) | \(172\) |
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Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.46 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=\frac {1}{27525120000} \, {\left (6967296000000 \, x^{7} + 30838579200000 \, x^{6} + 57746856960000 \, x^{5} + 58346097408000 \, x^{4} + 32457421737600 \, x^{3} + 6882844528480 \, x^{2} - 3991703112140 \, x - 5973304472091\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1988505613369}{26214400000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=-\frac {405}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} - \frac {49059}{448} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {739881}{3584} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {80346831}{358400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {4513921183}{28672000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {26326737569}{344064000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {16433930689}{65536000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1988505613369}{26214400000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {16433930689}{1310720000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (150) = 300\).
Time = 0.43 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.71 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=\frac {27}{458752000000} \, \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 {603}{71680000000} \, \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 {5769}{2560000000} \, \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 {30649}{320000000} \, \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 {1831}{300000} \, \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 {933}{5000} \, \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 {297}{125} \, \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 {216}{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. \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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