Optimal. Leaf size=158 \[ -\frac {184 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5 \sqrt {33}}+\frac {556 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}-\frac {92 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ \frac {556 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}-\frac {92 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {184 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {2}{3} \int \frac {90-103 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}-\frac {4}{99} \int \frac {\frac {7359}{2}-2277 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {8 \int \frac {47916+\frac {151371 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1089}\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {92}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {556}{5} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {184 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 100, normalized size = 0.63 \[ \frac {4}{15} \sqrt {2} \left (139 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-70 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {2 \sqrt {1-2 x} \left (4170 x^2+5144 x+1583\right )}{3 \sqrt {3 x+2} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{1125 \, x^{5} + 3525 \, x^{4} + 4415 \, x^{3} + 2763 \, x^{2} + 864 \, x + 108}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.39 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {3 x +2}\, \left (41700 x^{3}+30590 x^{2}-1390 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+700 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-9890 x -834 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+420 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-7915\right )}{15 \left (5 x +3\right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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