Optimal. Leaf size=172 \[ -\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac {3 (1-2 x)^{3/2} (1140 x+1963) (5 x+3)^{7/2}}{8000}-\frac {296633 (1-2 x)^{3/2} (5 x+3)^{5/2}}{128000}-\frac {3262963 (1-2 x)^{3/2} (5 x+3)^{3/2}}{307200}-\frac {35892593 (1-2 x)^{3/2} \sqrt {5 x+3}}{819200}+\frac {394818523 \sqrt {1-2 x} \sqrt {5 x+3}}{8192000}+\frac {4343003753 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8192000 \sqrt {10}} \]
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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} \[ -\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac {3 (1-2 x)^{3/2} (1140 x+1963) (5 x+3)^{7/2}}{8000}-\frac {296633 (1-2 x)^{3/2} (5 x+3)^{5/2}}{128000}-\frac {3262963 (1-2 x)^{3/2} (5 x+3)^{3/2}}{307200}-\frac {35892593 (1-2 x)^{3/2} \sqrt {5 x+3}}{819200}+\frac {394818523 \sqrt {1-2 x} \sqrt {5 x+3}}{8192000}+\frac {4343003753 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8192000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2} \, dx &=-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{70} \int \left (-385-\frac {1197 x}{2}\right ) \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {296633 \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx}{16000}\\ &=-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {3262963 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{51200}\\ &=-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {35892593 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{204800}\\ &=-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {394818523 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1638400}\\ &=\frac {394818523 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000}-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {4343003753 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16384000}\\ &=\frac {394818523 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000}-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {4343003753 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8192000 \sqrt {5}}\\ &=\frac {394818523 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000}-\frac {35892593 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200}-\frac {3262963 (1-2 x)^{3/2} (3+5 x)^{3/2}}{307200}-\frac {296633 (1-2 x)^{3/2} (3+5 x)^{5/2}}{128000}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (1963+1140 x)}{8000}+\frac {4343003753 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8192000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 89, normalized size = 0.52 \[ \frac {91203078813 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (33177600000 x^7+107550720000 x^6+127277568000 x^5+50509190400 x^4-23917446080 x^3-34142598520 x^2-20160334154 x+12531569067\right )}{1720320000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 87, normalized size = 0.51 \[ \frac {1}{172032000} \, {\left (16588800000 \, x^{6} + 62069760000 \, x^{5} + 94673664000 \, x^{4} + 72591427200 \, x^{3} + 24336990560 \, x^{2} - 4902803980 \, x - 12531569067\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {4343003753}{163840000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.48, size = 446, normalized size = 2.59 \[ \frac {9}{14336000000} \, \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 {171}{512000000} \, \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 {1353}{64000000} \, \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 {17119}{9600000} \, \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 {1353}{20000} \, \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 {513}{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 {108}{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 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (331776000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1241395200000 \sqrt {-10 x^{2}-x +3}\, x^{5}+1893473280000 \sqrt {-10 x^{2}-x +3}\, x^{4}+1451828544000 \sqrt {-10 x^{2}-x +3}\, x^{3}+486739811200 \sqrt {-10 x^{2}-x +3}\, x^{2}-98056079600 \sqrt {-10 x^{2}-x +3}\, x +91203078813 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-250631381340 \sqrt {-10 x^{2}-x +3}\right )}{3440640000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 121, normalized size = 0.70 \[ -\frac {135}{14} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {3933}{112} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {121887}{2240} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {8474351}{179200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {55355473}{2150400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {35892593}{409600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {4343003753}{163840000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {35892593}{8192000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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