Optimal. Leaf size=128 \[ -\frac {3 \sqrt {5 x+3} (11580 x+14629) (1-2 x)^{5/2}}{80000}-\frac {3}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {51373 \sqrt {5 x+3} (1-2 x)^{3/2}}{320000}+\frac {1695309 \sqrt {5 x+3} \sqrt {1-2 x}}{3200000}+\frac {18648399 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200000 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \begin {gather*} -\frac {3 \sqrt {5 x+3} (11580 x+14629) (1-2 x)^{5/2}}{80000}-\frac {3}{50} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {51373 \sqrt {5 x+3} (1-2 x)^{3/2}}{320000}+\frac {1695309 \sqrt {5 x+3} \sqrt {1-2 x}}{3200000}+\frac {18648399 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx &=-\frac {3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{50} \int \frac {\left (-179-\frac {579 x}{2}\right ) (1-2 x)^{3/2} (2+3 x)}{\sqrt {3+5 x}} \, dx\\ &=-\frac {3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {3 (1-2 x)^{5/2} \sqrt {3+5 x} (14629+11580 x)}{80000}+\frac {51373 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{32000}\\ &=\frac {51373 (1-2 x)^{3/2} \sqrt {3+5 x}}{320000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {3 (1-2 x)^{5/2} \sqrt {3+5 x} (14629+11580 x)}{80000}+\frac {1695309 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{640000}\\ &=\frac {1695309 \sqrt {1-2 x} \sqrt {3+5 x}}{3200000}+\frac {51373 (1-2 x)^{3/2} \sqrt {3+5 x}}{320000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {3 (1-2 x)^{5/2} \sqrt {3+5 x} (14629+11580 x)}{80000}+\frac {18648399 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{6400000}\\ &=\frac {1695309 \sqrt {1-2 x} \sqrt {3+5 x}}{3200000}+\frac {51373 (1-2 x)^{3/2} \sqrt {3+5 x}}{320000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {3 (1-2 x)^{5/2} \sqrt {3+5 x} (14629+11580 x)}{80000}+\frac {18648399 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3200000 \sqrt {5}}\\ &=\frac {1695309 \sqrt {1-2 x} \sqrt {3+5 x}}{3200000}+\frac {51373 (1-2 x)^{3/2} \sqrt {3+5 x}}{320000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {3 (1-2 x)^{5/2} \sqrt {3+5 x} (14629+11580 x)}{80000}+\frac {18648399 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 79, normalized size = 0.62 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (13824000 x^5+8812800 x^4-13767360 x^3-7793240 x^2+6001742 x-314441\right )+18648399 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{32000000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 141, normalized size = 1.10 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {95044375 (1-2 x)^4}{(5 x+3)^4}+\frac {169309500 (1-2 x)^3}{(5 x+3)^3}+\frac {88005120 (1-2 x)^2}{(5 x+3)^2}-\frac {28768880 (1-2 x)}{5 x+3}-2465904\right )}{3200000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {18648399 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{3200000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.67, size = 77, normalized size = 0.60 \begin {gather*} -\frac {1}{3200000} \, {\left (6912000 \, x^{4} + 7862400 \, x^{3} - 2952480 \, x^{2} - 5372860 \, x + 314441\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {18648399}{64000000} \, \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.43, size = 275, normalized size = 2.15 \begin {gather*} -\frac {9}{160000000} \, \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 {27}{3200000} \, \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 {3}{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 {1}{100} \, \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 {4}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 0.95 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}-157248000 \sqrt {-10 x^{2}-x +3}\, x^{3}+59049600 \sqrt {-10 x^{2}-x +3}\, x^{2}+107457200 \sqrt {-10 x^{2}-x +3}\, x +18648399 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-6288820 \sqrt {-10 x^{2}-x +3}\right )}{64000000 \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 92, normalized size = 0.72 \begin {gather*} -\frac {54}{25} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {2457}{1000} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + \frac {18453}{20000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {268643}{160000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {18648399}{64000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {314441}{3200000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3}{\sqrt {5\,x+3}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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