Optimal. Leaf size=172 \[ -\frac {3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac {255169 \sqrt {5 x+3} (1-2 x)^{7/2}}{640000}+\frac {2806859 \sqrt {5 x+3} (1-2 x)^{5/2}}{19200000}+\frac {30875449 \sqrt {5 x+3} (1-2 x)^{3/2}}{76800000}+\frac {339629939 \sqrt {5 x+3} \sqrt {1-2 x}}{256000000}+\frac {3735929329 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{256000000 \sqrt {10}} \]
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Rubi [A] time = 0.05, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac {3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac {255169 \sqrt {5 x+3} (1-2 x)^{7/2}}{640000}+\frac {2806859 \sqrt {5 x+3} (1-2 x)^{5/2}}{19200000}+\frac {30875449 \sqrt {5 x+3} (1-2 x)^{3/2}}{76800000}+\frac {339629939 \sqrt {5 x+3} \sqrt {1-2 x}}{256000000}+\frac {3735929329 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{256000000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx &=-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{70} \int \left (-253-\frac {801 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx\\ &=-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {255169 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{80000}\\ &=-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {2806859 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{1280000}\\ &=\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {30875449 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{7680000}\\ &=\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {339629939 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{51200000}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{512000000}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{256000000 \sqrt {5}}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 89, normalized size = 0.52 \[ \frac {78454515909 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (165888000000 x^7+111974400000 x^6-202397184000 x^5-117722284800 x^4+104908893760 x^3+49731440840 x^2-31177591322 x+679278531\right )}{53760000000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 87, normalized size = 0.51 \[ \frac {1}{5376000000} \, {\left (82944000000 \, x^{6} + 97459200000 \, x^{5} - 52468992000 \, x^{4} - 85095638400 \, x^{3} + 9906627680 \, x^{2} + 29819034260 \, x - 679278531\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3735929329}{5120000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.46, size = 446, normalized size = 2.59 \[ \frac {9}{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 {9}{400000000} \, \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 {33}{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 {17}{384000} \, \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 {77}{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 {13}{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 {12}{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 = 0.90 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (1658880000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1949184000000 \sqrt {-10 x^{2}-x +3}\, x^{5}-1049379840000 \sqrt {-10 x^{2}-x +3}\, x^{4}-1701912768000 \sqrt {-10 x^{2}-x +3}\, x^{3}+198132553600 \sqrt {-10 x^{2}-x +3}\, x^{2}+596380685200 \sqrt {-10 x^{2}-x +3}\, x +78454515909 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-13585570620 \sqrt {-10 x^{2}-x +3}\right )}{107520000000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 121, normalized size = 0.70 \[ -\frac {54}{35} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {1161}{700} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {47529}{70000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {5697497}{5600000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {5531929}{67200000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30875449}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3735929329}{5120000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30875449}{256000000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3} \,d x \]
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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