Optimal. Leaf size=249 \[ -\frac {923943703 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{691031250 \sqrt {33}}+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}{3575}+\frac {8318 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{482625}+\frac {25603 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{1876875}-\frac {6794792 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{84459375}-\frac {923943703 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{1520268750}-\frac {30660308017 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{691031250 \sqrt {33}} \]
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Rubi [A] time = 0.10, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}{3575}+\frac {8318 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{482625}+\frac {25603 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{1876875}-\frac {6794792 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{84459375}-\frac {923943703 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{1520268750}-\frac {923943703 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{691031250 \sqrt {33}}-\frac {30660308017 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{691031250 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx &=\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {2}{65} \int \left (-\frac {113}{2}-\frac {159 x}{2}\right ) (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx\\ &=\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {4 \int \left (-\frac {6063}{2}-\frac {12477 x}{4}\right ) \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx}{10725}\\ &=\frac {8318 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {8 \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (-\frac {826005}{8}+\frac {691281 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{1447875}\\ &=\frac {25603 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1876875}+\frac {8318 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {8 \int \frac {(3+5 x)^{3/2} \left (\frac {83150493}{16}+7644141 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{50675625}\\ &=-\frac {6794792 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{84459375}+\frac {25603 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1876875}+\frac {8318 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (-\frac {5392906641}{16}-\frac {8315493327 x}{16}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{760134375}\\ &=-\frac {923943703 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1520268750}-\frac {6794792 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{84459375}+\frac {25603 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1876875}+\frac {8318 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {8 \int \frac {\frac {349425411903}{32}+\frac {275942772153 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6841209375}\\ &=-\frac {923943703 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1520268750}-\frac {6794792 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{84459375}+\frac {25603 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1876875}+\frac {8318 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {923943703 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1382062500}+\frac {30660308017 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7601343750}\\ &=-\frac {923943703 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1520268750}-\frac {6794792 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{84459375}+\frac {25603 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1876875}+\frac {8318 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {30660308017 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{691031250 \sqrt {33}}-\frac {923943703 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{691031250 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 112, normalized size = 0.45 \[ \frac {-30830473835 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (14033250000 x^5+5400675000 x^4-13684072500 x^3-3707642250 x^2+5290733520 x+1020785999\right )+61320616034 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{22804031250 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 165, normalized size = 0.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (12629925000000 x^{8}+14543550000000 x^{7}-11536182000000 x^{6}-16439014800000 x^{5}+4104920740500 x^{4}+7811051450400 x^{3}+260663905110 x^{2}-1166697093390 x -61320616034 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+30830473835 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-183741479820\right )}{1368241875000 x^{3}+1048985437500 x^{2}-319256437500 x -273648375000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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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