Optimal. Leaf size=128 \[ \frac {1695309 \sqrt {1-2 x} \sqrt {3+5 x}}{3200000}+\frac {51373 (1-2 x)^{3/2} \sqrt {3+5 x}}{320000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {3 (1-2 x)^{5/2} \sqrt {3+5 x} (14629+11580 x)}{80000}+\frac {18648399 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200000 \sqrt {10}} \]
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Rubi [A]
time = 0.02, 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 = {102, 152, 52,
56, 222} \begin {gather*} \frac {18648399 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200000 \sqrt {10}}-\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 222
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 \text {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 = 0.22, size = 83, normalized size = 0.65 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (943323-14546375 x-35721740 x^2+8824800 x^3+60048000 x^4+34560000 x^5\right )-18648399 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{32000000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 121, normalized size = 0.95
method | result | size |
risch | \(\frac {\left (6912000 x^{4}+7862400 x^{3}-2952480 x^{2}-5372860 x +314441\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3200000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {18648399 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{64000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-138240000 x^{4} \sqrt {-10 x^{2}-x +3}-157248000 x^{3} \sqrt {-10 x^{2}-x +3}+59049600 x^{2} \sqrt {-10 x^{2}-x +3}+18648399 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+107457200 x \sqrt {-10 x^{2}-x +3}-6288820 \sqrt {-10 x^{2}-x +3}\right )}{64000000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, 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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Fricas [A]
time = 1.31, 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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Sympy [A]
time = 167.80, size = 675, normalized size = 5.27 \begin {gather*} - \frac {343 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} + \frac {441 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} - \frac {189 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{3872} + \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{3125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} + \frac {27 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} + \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{2662} + \frac {15 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{7744} + \frac {5 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{3748096} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {63 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{256}\right )}{15625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs.
\(2 (95) = 190\).
time = 1.05, 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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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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