Integrand size = 24, antiderivative size = 69 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x}}{3+5 x}-2 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {68 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {101, 162, 65, 212} \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=-2 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {68 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}}-\frac {\sqrt {1-2 x}}{5 x+3} \]
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Rule 65
Rule 101
Rule 162
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x}}{3+5 x}+\int \frac {-5+3 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx \\ & = -\frac {\sqrt {1-2 x}}{3+5 x}+21 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-34 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {\sqrt {1-2 x}}{3+5 x}-21 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+34 \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+5 x}-2 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {68 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x}}{3+5 x}-2 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {68 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \]
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Time = 0.99 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{55}-2 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}\) | \(54\) |
default | \(\frac {2 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{55}-2 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}\) | \(54\) |
risch | \(\frac {-1+2 x}{\left (3+5 x \right ) \sqrt {1-2 x}}-2 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}+\frac {68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{55}\) | \(58\) |
pseudoelliptic | \(\frac {-110 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right ) \sqrt {21}+68 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-55 \sqrt {1-2 x}}{165+275 x}\) | \(65\) |
trager | \(-\frac {\sqrt {1-2 x}}{3+5 x}-\frac {34 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{55}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )\) | \(106\) |
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Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=\frac {34 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, \sqrt {21} {\left (5 \, x + 3\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 55 \, \sqrt {-2 \, x + 1}}{55 \, {\left (5 \, x + 3\right )}} \]
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Time = 14.88 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.88 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=\sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right ) - \frac {7 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{11} - 44 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {34}{55} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {\sqrt {-2 \, x + 1}}{5 \, x + 3} \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {34}{55} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {\sqrt {-2 \, x + 1}}{5 \, x + 3} \]
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Time = 1.49 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx=\frac {68\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{55}-2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )-\frac {2\,\sqrt {1-2\,x}}{5\,\left (2\,x+\frac {6}{5}\right )} \]
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