Integrand size = 26, antiderivative size = 179 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800000 \sqrt {10}} \]
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Time = 0.04 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {102, 158, 152, 52, 56, 222} \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {70224051139 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800000 \sqrt {10}}-\frac {3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3-\frac {403 (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^2}{2800}-\frac {52760369 (1-2 x)^{3/2} (5 x+3)^{3/2}}{7680000}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)}{640000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {5 x+3}}{20480000}+\frac {6384004649 \sqrt {1-2 x} \sqrt {5 x+3}}{204800000} \]
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 158
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {1}{70} \int \left (-382-\frac {1209 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx \\ & = -\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}+\frac {\int \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2} \left (\frac {121905}{2}+\frac {382641 x}{4}\right ) \, dx}{4200} \\ & = -\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {52760369 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{1280000} \\ & = -\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {580364059 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{5120000} \\ & = -\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {6384004649 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{40960000} \\ & = \frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{409600000} \\ & = \frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{204800000 \sqrt {5}} \\ & = \frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800000 \sqrt {10}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.52 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-604565196363-1226406863625 x+843850516940 x^2+5499280071200 x^3+10250528112000 x^4+10254608640000 x^5+5498496000000 x^6+1244160000000 x^7\right )-1474705073919 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{43008000000 \sqrt {3+5 x}} \]
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Time = 1.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {\left (248832000000 x^{6}+950400000000 x^{5}+1480681728000 x^{4}+1161696585600 x^{3}+402838062880 x^{2}-72932734340 x -201521732121\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4300800000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {70224051139 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4096000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(118\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4976640000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+19008000000000 x^{5} \sqrt {-10 x^{2}-x +3}+29613634560000 x^{4} \sqrt {-10 x^{2}-x +3}+23233931712000 x^{3} \sqrt {-10 x^{2}-x +3}+8056761257600 x^{2} \sqrt {-10 x^{2}-x +3}+1474705073919 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1458654686800 x \sqrt {-10 x^{2}-x +3}-4030434642420 \sqrt {-10 x^{2}-x +3}\right )}{86016000000 \sqrt {-10 x^{2}-x +3}}\) | \(155\) |
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Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.49 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {1}{4300800000} \, {\left (248832000000 \, x^{6} + 950400000000 \, x^{5} + 1480681728000 \, x^{4} + 1161696585600 \, x^{3} + 402838062880 \, x^{2} - 72932734340 \, x - 201521732121\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {70224051139}{4096000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=-\frac {81}{14} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {12051}{560} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {1904661}{56000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {134695173}{4480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {890455739}{53760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {580364059}{10240000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {70224051139}{4096000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {580364059}{204800000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (134) = 268\).
Time = 0.37 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.49 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {27}{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 {261}{1280000000} \, \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 {4203}{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 {451}{400000} \, \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 {653}{15000} \, \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 {84}{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 {72}{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. \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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