Optimal. Leaf size=249 \[ -\frac {70536439 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{13820625 \sqrt {33}}+\frac {2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac {67 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{160875}-\frac {160084 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{3378375}-\frac {2133359 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{6756750}-\frac {70536439 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{30405375}-\frac {9380126059 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{55282500 \sqrt {33}} \]
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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} \[ \frac {2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac {67 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{160875}-\frac {160084 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{3378375}-\frac {2133359 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{6756750}-\frac {70536439 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{30405375}-\frac {70536439 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{13820625 \sqrt {33}}-\frac {9380126059 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{55282500 \sqrt {33}} \]
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)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx &=\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {2}{65} \int \left (-\frac {69}{2}-\frac {93 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx\\ &=\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-870-\frac {201 x}{4}\right ) \sqrt {2+3 x} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx}{10725}\\ &=-\frac {67 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{160875}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {4 \int \frac {(3+5 x)^{5/2} \left (\frac {639867}{8}+120063 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{482625}\\ &=-\frac {160084 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{3378375}-\frac {67 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{160875}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {62883465}{8}-\frac {96001155 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{10135125}\\ &=-\frac {2133359 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{6756750}-\frac {160084 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{3378375}-\frac {67 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{160875}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {4 \int \frac {\sqrt {3+5 x} \left (\frac {8251543035}{16}+\frac {3174139755 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{152026875}\\ &=-\frac {70536439 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{30405375}-\frac {2133359 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{6756750}-\frac {160084 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{3378375}-\frac {67 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{160875}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {4 \int \frac {-\frac {267229618515}{16}-\frac {422105672655 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1368241875}\\ &=-\frac {70536439 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{30405375}-\frac {2133359 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{6756750}-\frac {160084 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{3378375}-\frac {67 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{160875}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {70536439 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{27641250}+\frac {9380126059 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{608107500}\\ &=-\frac {70536439 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{30405375}-\frac {2133359 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{6756750}-\frac {160084 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{3378375}-\frac {67 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{160875}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {9380126059 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{55282500 \sqrt {33}}-\frac {70536439 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{13820625 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 115, normalized size = 0.46 \[ \frac {9380126059 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (944944217 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (1403325000 x^5+2364390000 x^4+496455750 x^3-1110242250 x^2-638983395 x+67302101\right )\right )}{912161250 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-1262992500000 x^{8}-3096245250000 x^{7}-1783541025000 x^{6}+1405783957500 x^{5}+1870998115500 x^{4}+236537814150 x^{3}-380468567640 x^{2}-100883569890 x -9380126059 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4724721085 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+12114378180\right )}{54729675000 x^{3}+41959417500 x^{2}-12770257500 x -10945935000} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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