Optimal. Leaf size=179 \[ -\frac {3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3-\frac {403 (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^2}{2800}-\frac {52760369 (1-2 x)^{3/2} (5 x+3)^{3/2}}{7680000}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)}{640000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {5 x+3}}{20480000}+\frac {6384004649 \sqrt {1-2 x} \sqrt {5 x+3}}{204800000}+\frac {70224051139 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800000 \sqrt {10}} \]
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Rubi [A] time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 153, 147, 50, 54, 216} \[ -\frac {3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3-\frac {403 (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^2}{2800}-\frac {52760369 (1-2 x)^{3/2} (5 x+3)^{3/2}}{7680000}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)}{640000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {5 x+3}}{20480000}+\frac {6384004649 \sqrt {1-2 x} \sqrt {5 x+3}}{204800000}+\frac {70224051139 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx &=-\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 \operatorname {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*}
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Mathematica [A] time = 0.12, size = 89, normalized size = 0.50 \[ \frac {1474705073919 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (497664000000 x^7+1651968000000 x^6+2010963456000 x^5+842711443200 x^4-356020459840 x^3-548703531560 x^2-330110729902 x+201521732121\right )}{43008000000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 87, normalized size = 0.49 \[ \frac {1}{4300800000} \, {\left (248832000000 \, x^{6} + 950400000000 \, x^{5} + 1480681728000 \, x^{4} + 1161696585600 \, x^{3} + 402838062880 \, x^{2} - 72932734340 \, x - 201521732121\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {70224051139}{4096000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.50, size = 446, normalized size = 2.49 \[ \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 )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 155, normalized size = 0.87 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (4976640000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+19008000000000 \sqrt {-10 x^{2}-x +3}\, x^{5}+29613634560000 \sqrt {-10 x^{2}-x +3}\, x^{4}+23233931712000 \sqrt {-10 x^{2}-x +3}\, x^{3}+8056761257600 \sqrt {-10 x^{2}-x +3}\, x^{2}-1458654686800 \sqrt {-10 x^{2}-x +3}\, x +1474705073919 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-4030434642420 \sqrt {-10 x^{2}-x +3}\right )}{86016000000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 121, normalized size = 0.68 \[ -\frac {81}{14} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {12051}{560} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {1904661}{56000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {134695173}{4480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {890455739}{53760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {580364059}{10240000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {70224051139}{4096000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {580364059}{204800000} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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