Optimal. Leaf size=89 \[ \frac {560}{81} \sqrt {1-2 x}+\frac {80}{81} (1-2 x)^{3/2}+\frac {16}{63} (1-2 x)^{5/2}+\frac {(1-2 x)^{7/2}}{21 (2+3 x)}-\frac {560}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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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 = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 52, 65, 212}
\begin {gather*} \frac {(1-2 x)^{7/2}}{21 (3 x+2)}+\frac {16}{63} (1-2 x)^{5/2}+\frac {80}{81} (1-2 x)^{3/2}+\frac {560}{81} \sqrt {1-2 x}-\frac {560}{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 79
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx &=\frac {(1-2 x)^{7/2}}{21 (2+3 x)}+\frac {40}{21} \int \frac {(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=\frac {16}{63} (1-2 x)^{5/2}+\frac {(1-2 x)^{7/2}}{21 (2+3 x)}+\frac {40}{9} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {80}{81} (1-2 x)^{3/2}+\frac {16}{63} (1-2 x)^{5/2}+\frac {(1-2 x)^{7/2}}{21 (2+3 x)}+\frac {280}{27} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {560}{81} \sqrt {1-2 x}+\frac {80}{81} (1-2 x)^{3/2}+\frac {16}{63} (1-2 x)^{5/2}+\frac {(1-2 x)^{7/2}}{21 (2+3 x)}+\frac {1960}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {560}{81} \sqrt {1-2 x}+\frac {80}{81} (1-2 x)^{3/2}+\frac {16}{63} (1-2 x)^{5/2}+\frac {(1-2 x)^{7/2}}{21 (2+3 x)}-\frac {1960}{81} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {560}{81} \sqrt {1-2 x}+\frac {80}{81} (1-2 x)^{3/2}+\frac {16}{63} (1-2 x)^{5/2}+\frac {(1-2 x)^{7/2}}{21 (2+3 x)}-\frac {560}{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 = 0.12, size = 63, normalized size = 0.71 \begin {gather*} \frac {1}{243} \left (\frac {3 \sqrt {1-2 x} \left (1325+1474 x-516 x^2+216 x^3\right )}{2+3 x}-560 \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 = 0.10, size = 63, normalized size = 0.71
method | result | size |
risch | \(-\frac {432 x^{4}-1248 x^{3}+3464 x^{2}+1176 x -1325}{81 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {560 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(56\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{9}+\frac {74 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {182 \sqrt {1-2 x}}{27}-\frac {98 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}-\frac {560 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{9}+\frac {74 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {182 \sqrt {1-2 x}}{27}-\frac {98 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}-\frac {560 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
trager | \(\frac {\left (216 x^{3}-516 x^{2}+1474 x +1325\right ) \sqrt {1-2 x}}{162+243 x}+\frac {280 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{243}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 80, normalized size = 0.90 \begin {gather*} \frac {2}{9} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {74}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {280}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {182}{27} \, \sqrt {-2 \, x + 1} + \frac {49 \, \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 = 1.81, size = 75, normalized size = 0.84 \begin {gather*} \frac {280 \, \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 (216 \, x^{3} - 516 \, x^{2} + 1474 \, x + 1325\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 = 0.56, size = 90, normalized size = 1.01 \begin {gather*} \frac {2}{9} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {74}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {280}{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 {182}{27} \, \sqrt {-2 \, x + 1} + \frac {49 \, \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 = 0.06, size = 64, normalized size = 0.72 \begin {gather*} \frac {98\,\sqrt {1-2\,x}}{243\,\left (2\,x+\frac {4}{3}\right )}+\frac {182\,\sqrt {1-2\,x}}{27}+\frac {74\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {2\,{\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 )\,560{}\mathrm {i}}{243} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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