Optimal. Leaf size=172 \[ \frac {565404807 \sqrt {1-2 x} \sqrt {3+5 x}}{102400000}+\frac {17133479 (1-2 x)^{3/2} \sqrt {3+5 x}}{10240000}-\frac {1557589 (1-2 x)^{5/2} \sqrt {3+5 x}}{512000}-\frac {141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {6219452877 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400000 \sqrt {10}} \]
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Rubi [A]
time = 0.04, 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 = {102, 152, 52,
56, 222} \begin {gather*} \frac {6219452877 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400000 \sqrt {10}}-\frac {3}{70} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {141599 (5 x+3)^{3/2} (1-2 x)^{5/2}}{128000}-\frac {3 (5 x+3)^{5/2} (33300 x+49829) (1-2 x)^{5/2}}{280000}-\frac {1557589 \sqrt {5 x+3} (1-2 x)^{5/2}}{512000}+\frac {17133479 \sqrt {5 x+3} (1-2 x)^{3/2}}{10240000}+\frac {565404807 \sqrt {5 x+3} \sqrt {1-2 x}}{102400000} \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 (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx &=-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{70} \int \left (-319-\frac {999 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx\\ &=-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {141599 \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx}{16000}\\ &=-\frac {141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {4672767 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{256000}\\ &=-\frac {1557589 (1-2 x)^{5/2} \sqrt {3+5 x}}{512000}-\frac {141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {17133479 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{1024000}\\ &=\frac {17133479 (1-2 x)^{3/2} \sqrt {3+5 x}}{10240000}-\frac {1557589 (1-2 x)^{5/2} \sqrt {3+5 x}}{512000}-\frac {141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {565404807 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{20480000}\\ &=\frac {565404807 \sqrt {1-2 x} \sqrt {3+5 x}}{102400000}+\frac {17133479 (1-2 x)^{3/2} \sqrt {3+5 x}}{10240000}-\frac {1557589 (1-2 x)^{5/2} \sqrt {3+5 x}}{512000}-\frac {141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {6219452877 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{204800000}\\ &=\frac {565404807 \sqrt {1-2 x} \sqrt {3+5 x}}{102400000}+\frac {17133479 (1-2 x)^{3/2} \sqrt {3+5 x}}{10240000}-\frac {1557589 (1-2 x)^{5/2} \sqrt {3+5 x}}{512000}-\frac {141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {6219452877 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{102400000 \sqrt {5}}\\ &=\frac {565404807 \sqrt {1-2 x} \sqrt {3+5 x}}{102400000}+\frac {17133479 (1-2 x)^{3/2} \sqrt {3+5 x}}{10240000}-\frac {1557589 (1-2 x)^{5/2} \sqrt {3+5 x}}{512000}-\frac {141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac {6219452877 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 93, normalized size = 0.54 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (11857233303-8103253875 x-132564854140 x^2-181119127200 x^3+77560528000 x^4+437564160000 x^5+422784000000 x^6+138240000000 x^7\right )-43536170139 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{7168000000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 155, normalized size = 0.90
method | result | size |
risch | \(\frac {\left (27648000000 x^{6}+67968000000 x^{5}+46732032000 x^{4}-12527113600 x^{3}-28707557280 x^{2}-9288436460 x +3952411101\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{716800000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {6219452877 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2048000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(118\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-552960000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-1359360000000 x^{5} \sqrt {-10 x^{2}-x +3}-934640640000 x^{4} \sqrt {-10 x^{2}-x +3}+250542272000 x^{3} \sqrt {-10 x^{2}-x +3}+574151145600 x^{2} \sqrt {-10 x^{2}-x +3}+43536170139 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+185768729200 x \sqrt {-10 x^{2}-x +3}-79048222020 \sqrt {-10 x^{2}-x +3}\right )}{14336000000 \sqrt {-10 x^{2}-x +3}}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 116, normalized size = 0.67 \begin {gather*} -\frac {27}{70} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} - \frac {2439}{2800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {197487}{280000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {141599}{64000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {141599}{1280000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {51400437}{5120000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {6219452877}{2048000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {51400437}{102400000} \, \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 = 0.68, size = 87, normalized size = 0.51 \begin {gather*} -\frac {1}{716800000} \, {\left (27648000000 \, x^{6} + 67968000000 \, x^{5} + 46732032000 \, x^{4} - 12527113600 \, x^{3} - 28707557280 \, x^{2} - 9288436460 \, x + 3952411101\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {6219452877}{2048000000} \, \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 [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 446 vs.
\(2 (127) = 254\).
time = 1.75, size = 446, normalized size = 2.59 \begin {gather*} -\frac {9}{35840000000} \, \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 {243}{2560000000} \, \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 {561}{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 {769}{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 {319}{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 {21}{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 {36}{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 {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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