Optimal. Leaf size=113 \[ \frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {1207 \sqrt {1-2 x}}{49 (2+3 x)}+\frac {83264 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {101, 156, 162,
65, 212} \begin {gather*} \frac {1207 \sqrt {1-2 x}}{49 (3 x+2)}+\frac {52 \sqrt {1-2 x}}{21 (3 x+2)^2}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}+\frac {83264 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \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 101
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}-\frac {1}{3} \int \frac {-18+25 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}-\frac {1}{42} \int \frac {-1374+1560 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {1207 \sqrt {1-2 x}}{49 (2+3 x)}-\frac {1}{294} \int \frac {-59124+36210 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {1207 \sqrt {1-2 x}}{49 (2+3 x)}-\frac {41632}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+1375 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {1207 \sqrt {1-2 x}}{49 (2+3 x)}+\frac {41632}{49} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-1375 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {1207 \sqrt {1-2 x}}{49 (2+3 x)}+\frac {83264 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 83, normalized size = 0.73 \begin {gather*} \frac {\sqrt {1-2 x} \left (5087+14848 x+10863 x^2\right )}{49 (2+3 x)^3}+\frac {83264 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 75, normalized size = 0.66
method | result | size |
risch | \(-\frac {21726 x^{3}+18833 x^{2}-4674 x -5087}{49 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {83264 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(69\) |
derivativedivides | \(-50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1207 \left (1-2 x \right )^{\frac {5}{2}}}{147}-\frac {7346 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {1243 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {83264 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(75\) |
default | \(-50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1207 \left (1-2 x \right )^{\frac {5}{2}}}{147}-\frac {7346 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {1243 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {83264 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(75\) |
trager | \(\frac {\left (10863 x^{2}+14848 x +5087\right ) \sqrt {1-2 x}}{49 \left (2+3 x \right )^{3}}-\frac {41632 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{1029}-25 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 128, normalized size = 1.13 \begin {gather*} 25 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {41632}{1029} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (10863 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 51422 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 60907 \, \sqrt {-2 \, x + 1}\right )}}{49 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.97, size = 130, normalized size = 1.15 \begin {gather*} \frac {25725 \, \sqrt {55} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 41632 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10863 \, x^{2} + 14848 \, x + 5087\right )} \sqrt {-2 \, x + 1}}{1029 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 115.72, size = 614, normalized size = 5.43 \begin {gather*} 660 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 264 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) + 112 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 1650 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right ) + 2750 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.48, size = 123, normalized size = 1.09 \begin {gather*} 25 \, \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 {41632}{1029} \, \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 {10863 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 51422 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 60907 \, \sqrt {-2 \, x + 1}}{196 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 89, normalized size = 0.79 \begin {gather*} \frac {83264\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029}-50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {2486\,\sqrt {1-2\,x}}{27}-\frac {14692\,{\left (1-2\,x\right )}^{3/2}}{189}+\frac {2414\,{\left (1-2\,x\right )}^{5/2}}{147}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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