Optimal. Leaf size=95 \[ \frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac {49 \sqrt {1-2 x}}{2 (2+3 x)}+235 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 95, 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, 154, 162,
65, 212} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac {49 \sqrt {1-2 x}}{2 (3 x+2)}+235 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-242 \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 154
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)} \, dx &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {(129-27 x) \sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac {49 \sqrt {1-2 x}}{2 (2+3 x)}-\frac {1}{18} \int \frac {-3501+2151 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac {49 \sqrt {1-2 x}}{2 (2+3 x)}-\frac {1645}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+1331 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac {49 \sqrt {1-2 x}}{2 (2+3 x)}+\frac {1645}{2} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-1331 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac {49 \sqrt {1-2 x}}{2 (2+3 x)}+235 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-242 \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 = 80, normalized size = 0.84 \begin {gather*} \frac {7 \sqrt {1-2 x} (43+61 x)}{6 (2+3 x)^2}+235 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-242 \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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Maple [A]
time = 0.16, size = 66, normalized size = 0.69
method | result | size |
risch | \(-\frac {7 \left (122 x^{2}+25 x -43\right )}{6 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}+\frac {235 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(64\) |
derivativedivides | \(-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}-\frac {126 \left (\frac {61 \left (1-2 x \right )^{\frac {3}{2}}}{54}-\frac {49 \sqrt {1-2 x}}{18}\right )}{\left (-4-6 x \right )^{2}}+\frac {235 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(66\) |
default | \(-\frac {242 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}-\frac {126 \left (\frac {61 \left (1-2 x \right )^{\frac {3}{2}}}{54}-\frac {49 \sqrt {1-2 x}}{18}\right )}{\left (-4-6 x \right )^{2}}+\frac {235 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(66\) |
trager | \(\frac {7 \left (61 x +43\right ) \sqrt {1-2 x}}{6 \left (2+3 x \right )^{2}}+\frac {121 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{5}-\frac {235 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{6}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 110, normalized size = 1.16 \begin {gather*} \frac {121}{5} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {235}{6} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {7 \, {\left (61 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 147 \, \sqrt {-2 \, x + 1}\right )}}{3 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.26, size = 122, normalized size = 1.28 \begin {gather*} \frac {726 \, \sqrt {11} \sqrt {5} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 1175 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 35 \, {\left (61 \, x + 43\right )} \sqrt {-2 \, x + 1}}{30 \, {\left (9 \, x^{2} + 12 \, x + 4\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 = 0.98, size = 107, normalized size = 1.13 \begin {gather*} \frac {121}{5} \, \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 {235}{6} \, \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 {7 \, {\left (61 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 147 \, \sqrt {-2 \, x + 1}\right )}}{12 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 71, normalized size = 0.75 \begin {gather*} \frac {235\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3}-\frac {242\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{5}+\frac {\frac {343\,\sqrt {1-2\,x}}{9}-\frac {427\,{\left (1-2\,x\right )}^{3/2}}{27}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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