Optimal. Leaf size=138 \[ -\frac {3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {407}{640} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {4477 \sqrt {5 x+3} (1-2 x)^{3/2}}{12800}+\frac {147741 \sqrt {5 x+3} \sqrt {1-2 x}}{128000}+\frac {1625151 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac {3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {407}{640} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {4477 \sqrt {5 x+3} (1-2 x)^{3/2}}{12800}+\frac {147741 \sqrt {5 x+3} \sqrt {1-2 x}}{128000}+\frac {1625151 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx &=-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {37}{20} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1221}{320} \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx\\ &=-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {4477 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{1280}\\ &=\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {147741 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {147741 \sqrt {1-2 x} \sqrt {3+5 x}}{128000}+\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1625151 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{256000}\\ &=\frac {147741 \sqrt {1-2 x} \sqrt {3+5 x}}{128000}+\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1625151 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{128000 \sqrt {5}}\\ &=\frac {147741 \sqrt {1-2 x} \sqrt {3+5 x}}{128000}+\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1625151 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.57 \[ \frac {10 \sqrt {5 x+3} \left (1536000 x^5+723200 x^4-1474240 x^3-614360 x^2+582958 x-46809\right )+1625151 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1280000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 77, normalized size = 0.56 \[ -\frac {1}{128000} \, {\left (768000 \, x^{4} + 745600 \, x^{3} - 364320 \, x^{2} - 489340 \, x + 46809\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1625151}{2560000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.38, size = 275, normalized size = 1.99 \[ -\frac {1}{6400000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {41}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {17}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {51}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 0.88 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-15360000 \sqrt {-10 x^{2}-x +3}\, x^{4}-14912000 \sqrt {-10 x^{2}-x +3}\, x^{3}+7286400 \sqrt {-10 x^{2}-x +3}\, x^{2}+9786800 \sqrt {-10 x^{2}-x +3}\, x +1625151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-936180 \sqrt {-10 x^{2}-x +3}\right )}{2560000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 84, normalized size = 0.61 \[ -\frac {3}{50} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {37}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {37}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {13431}{6400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1625151}{2560000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {13431}{128000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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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