Optimal. Leaf size=89 \[ -\frac {146}{81} \sqrt {1-2 x}-\frac {146}{567} (1-2 x)^{3/2}-\frac {5}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}+\frac {146}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 89, 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 = {91, 81, 52, 65,
212} \begin {gather*} -\frac {(1-2 x)^{5/2}}{63 (3 x+2)}-\frac {5}{9} (1-2 x)^{5/2}-\frac {146}{567} (1-2 x)^{3/2}-\frac {146}{81} \sqrt {1-2 x}+\frac {146}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}+\frac {1}{63} \int \frac {(1-2 x)^{3/2} (277+525 x)}{2+3 x} \, dx\\ &=-\frac {5}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}-\frac {73}{63} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {146}{567} (1-2 x)^{3/2}-\frac {5}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}-\frac {73}{27} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {146}{81} \sqrt {1-2 x}-\frac {146}{567} (1-2 x)^{3/2}-\frac {5}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}-\frac {511}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {146}{81} \sqrt {1-2 x}-\frac {146}{567} (1-2 x)^{3/2}-\frac {5}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}+\frac {511}{81} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {146}{81} \sqrt {1-2 x}-\frac {146}{567} (1-2 x)^{3/2}-\frac {5}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}+\frac {146}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 63, normalized size = 0.71 \begin {gather*} \frac {1}{243} \left (-\frac {3 \sqrt {1-2 x} \left (425+187 x-300 x^2+540 x^3\right )}{2+3 x}+146 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 63, normalized size = 0.71
method | result | size |
risch | \(\frac {1080 x^{4}-1140 x^{3}+674 x^{2}+663 x -425}{81 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {146 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(56\) |
derivativedivides | \(-\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{9}-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {16 \sqrt {1-2 x}}{9}+\frac {14 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}+\frac {146 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
default | \(-\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{9}-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {16 \sqrt {1-2 x}}{9}+\frac {14 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}+\frac {146 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
trager | \(-\frac {\left (540 x^{3}-300 x^{2}+187 x +425\right ) \sqrt {1-2 x}}{81 \left (2+3 x \right )}+\frac {73 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{243}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 80, normalized size = 0.90 \begin {gather*} -\frac {5}{9} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {73}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {16}{9} \, \sqrt {-2 \, x + 1} - \frac {7 \, \sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 76, normalized size = 0.85 \begin {gather*} \frac {73 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - 3 \, {\left (540 \, x^{3} - 300 \, x^{2} + 187 \, x + 425\right )} \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 90, normalized size = 1.01 \begin {gather*} -\frac {5}{9} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {73}{243} \, \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 {16}{9} \, \sqrt {-2 \, x + 1} - \frac {7 \, \sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 64, normalized size = 0.72 \begin {gather*} -\frac {14\,\sqrt {1-2\,x}}{243\,\left (2\,x+\frac {4}{3}\right )}-\frac {16\,\sqrt {1-2\,x}}{9}-\frac {20\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {5\,{\left (1-2\,x\right )}^{5/2}}{9}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,146{}\mathrm {i}}{243} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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