Optimal. Leaf size=280 \[ -\frac {50299451003 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{4146187500 \sqrt {33}}+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac {178 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac {2503 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac {199721 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{12065625}-\frac {57509209 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{506756250}-\frac {380132617 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{506756250}-\frac {50299451003 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{9121612500}-\frac {836091184171 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2073093750 \sqrt {33}} \]
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Rubi [A] time = 0.12, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac {178 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac {2503 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac {199721 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{12065625}-\frac {57509209 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{506756250}-\frac {380132617 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{506756250}-\frac {50299451003 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{9121612500}-\frac {50299451003 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4146187500 \sqrt {33}}-\frac {836091184171 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2073093750 \sqrt {33}} \]
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)^{5/2} (3+5 x)^{5/2} \, dx &=\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac {2}{75} \int \left (-\frac {71}{2}-\frac {89 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac {4 \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2} \left (-\frac {2339}{2}+\frac {2503 x}{4}\right )}{\sqrt {1-2 x}} \, dx}{14625}\\ &=\frac {2503 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac {4 \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2} \left (\frac {816405}{8}+\frac {599163 x}{4}\right )}{\sqrt {1-2 x}} \, dx}{804375}\\ &=-\frac {199721 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{12065625}+\frac {2503 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {113620371}{8}-\frac {172527627 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{36196875}\\ &=-\frac {57509209 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{506756250}-\frac {199721 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{12065625}+\frac {2503 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac {4 \int \frac {(3+5 x)^{3/2} \left (\frac {22424965215}{16}+\frac {17105967765 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{760134375}\\ &=-\frac {380132617 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{506756250}-\frac {57509209 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{506756250}-\frac {199721 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{12065625}+\frac {2503 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {735492282165}{8}-\frac {2263475295135 x}{16}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{11402015625}\\ &=-\frac {50299451003 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{9121612500}-\frac {380132617 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{506756250}-\frac {57509209 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{506756250}-\frac {199721 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{12065625}+\frac {2503 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac {4 \int \frac {\frac {95277493539765}{32}+\frac {37624103287695 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{102618140625}\\ &=-\frac {50299451003 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{9121612500}-\frac {380132617 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{506756250}-\frac {57509209 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{506756250}-\frac {199721 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{12065625}+\frac {2503 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac {50299451003 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8292375000}+\frac {836091184171 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{22804031250}\\ &=-\frac {50299451003 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{9121612500}-\frac {380132617 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{506756250}-\frac {57509209 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{506756250}-\frac {199721 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{12065625}+\frac {2503 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac {836091184171 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2073093750 \sqrt {33}}-\frac {50299451003 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4146187500 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 119, normalized size = 0.42 \[ \frac {\sqrt {2} \left (3344364736684 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-1684482853585 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )-30 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (547296750000 x^6+1316318850000 x^5+888419542500 x^4-227285730000 x^3-522917547750 x^2-177853891770 x+44426819351\right )}{273648375000} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 170, normalized size = 0.61 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-492567075000000 x^{9}-1562321722500000 x^{8}-1592905277250000 x^{7}-33511953825000 x^{6}+1050958443600000 x^{5}+633067124890500 x^{4}-67989068522100 x^{3}-162128981218890 x^{2}-22684068454890 x -3344364736684 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1684482853585 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+7996827483180\right )}{8209451250000 x^{3}+6293912625000 x^{2}-1915538625000 x -1641890250000} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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