Optimal. Leaf size=92 \[ -\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {836}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 159, 162,
65, 212} \begin {gather*} -\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}-\frac {58}{75} \sqrt {1-2 x}-\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {836}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 159
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx &=-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {1}{5} \int \frac {\sqrt {1-2 x} (101+29 x)}{(2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {2}{75} \int \frac {\frac {1863}{2}-\frac {1493 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}+\frac {343}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {4598}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {343}{3} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {4598}{25} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {836}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 83, normalized size = 0.90 \begin {gather*} -\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1}{375} \left (\frac {5 \sqrt {1-2 x} (-339+40 x)}{3+5 x}+2508 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 63, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {8 \sqrt {1-2 x}}{75}+\frac {242 \sqrt {1-2 x}}{125 \left (-\frac {6}{5}-2 x \right )}+\frac {836 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}-\frac {98 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\) | \(63\) |
default | \(\frac {8 \sqrt {1-2 x}}{75}+\frac {242 \sqrt {1-2 x}}{125 \left (-\frac {6}{5}-2 x \right )}+\frac {836 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}-\frac {98 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\) | \(63\) |
risch | \(-\frac {80 x^{2}-718 x +339}{75 \left (3+5 x \right ) \sqrt {1-2 x}}+\frac {836 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}-\frac {98 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\) | \(64\) |
trager | \(\frac {\left (-339+40 x \right ) \sqrt {1-2 x}}{225+375 x}-\frac {49 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{9}-\frac {418 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{125}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 98, normalized size = 1.07 \begin {gather*} -\frac {418}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {49}{9} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8}{75} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 107, normalized size = 1.16 \begin {gather*} \frac {3762 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 6125 \, \sqrt {7} \sqrt {3} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 15 \, {\left (40 \, x - 339\right )} \sqrt {-2 \, x + 1}}{1125 \, {\left (5 \, x + 3\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 = 0.70, size = 104, normalized size = 1.13 \begin {gather*} -\frac {418}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {49}{9} \, \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 {8}{75} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 66, normalized size = 0.72 \begin {gather*} \frac {8\,\sqrt {1-2\,x}}{75}-\frac {242\,\sqrt {1-2\,x}}{125\,\left (2\,x+\frac {6}{5}\right )}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{9}-\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,836{}\mathrm {i}}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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