Optimal. Leaf size=94 \[ \frac {47 (1-2 x)^{5/2}}{294 (3 x+2)}-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}+\frac {2873 (1-2 x)^{3/2}}{3969}+\frac {2873}{567} \sqrt {1-2 x}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 78, 50, 63, 206} \[ \frac {47 (1-2 x)^{5/2}}{294 (3 x+2)}-\frac {(1-2 x)^{5/2}}{126 (3 x+2)^2}+\frac {2873 (1-2 x)^{3/2}}{3969}+\frac {2873}{567} \sqrt {1-2 x}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {(1-2 x)^{3/2} (559+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac {47 (1-2 x)^{5/2}}{294 (2+3 x)}+\frac {2873}{882} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {2873 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac {47 (1-2 x)^{5/2}}{294 (2+3 x)}+\frac {2873}{378} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {2873}{567} \sqrt {1-2 x}+\frac {2873 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac {47 (1-2 x)^{5/2}}{294 (2+3 x)}+\frac {2873}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {2873}{567} \sqrt {1-2 x}+\frac {2873 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac {47 (1-2 x)^{5/2}}{294 (2+3 x)}-\frac {2873}{162} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2873}{567} \sqrt {1-2 x}+\frac {2873 (1-2 x)^{3/2}}{3969}-\frac {(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac {47 (1-2 x)^{5/2}}{294 (2+3 x)}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.67 \[ \frac {\sqrt {1-2 x} \left (-1800 x^3+5520 x^2+10195 x+3803\right )}{162 (3 x+2)^2}-\frac {2873 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 79, normalized size = 0.84 \[ \frac {2873 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1800 \, x^{3} - 5520 \, x^{2} - 10195 \, x - 3803\right )} \sqrt {-2 \, x + 1}}{3402 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 86, normalized size = 0.91 \[ \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {2873}{3402} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {130}{27} \, \sqrt {-2 \, x + 1} - \frac {435 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1001 \, \sqrt {-2 \, x + 1}}{324 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.70 \[ -\frac {2873 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1701}+\frac {50 \left (-2 x +1\right )^{\frac {3}{2}}}{81}+\frac {130 \sqrt {-2 x +1}}{27}+\frac {-\frac {145 \left (-2 x +1\right )^{\frac {3}{2}}}{27}+\frac {1001 \sqrt {-2 x +1}}{81}}{\left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 92, normalized size = 0.98 \[ \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {2873}{3402} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {130}{27} \, \sqrt {-2 \, x + 1} - \frac {435 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1001 \, \sqrt {-2 \, x + 1}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 73, normalized size = 0.78 \[ \frac {130\,\sqrt {1-2\,x}}{27}+\frac {50\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {\frac {1001\,\sqrt {1-2\,x}}{729}-\frac {145\,{\left (1-2\,x\right )}^{3/2}}{243}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,2873{}\mathrm {i}}{1701} \]
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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