Integrand size = 17, antiderivative size = 48 \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x}}{5 (3+5 x)}+\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {43, 65, 212} \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}-\frac {\sqrt {1-2 x}}{5 (5 x+3)} \]
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Rule 43
Rule 65
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x}}{5 (3+5 x)}-\frac {1}{5} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {\sqrt {1-2 x}}{5 (3+5 x)}+\frac {1}{5} \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}}{5 (3+5 x)}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x}}{15+25 x}+\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}} \]
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Time = 0.96 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{25 \left (-\frac {6}{5}-2 x \right )}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{275}\) | \(36\) |
default | \(\frac {2 \sqrt {1-2 x}}{25 \left (-\frac {6}{5}-2 x \right )}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{275}\) | \(36\) |
risch | \(\frac {-1+2 x}{5 \left (3+5 x \right ) \sqrt {1-2 x}}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{275}\) | \(41\) |
pseudoelliptic | \(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-55 \sqrt {1-2 x}}{825+1375 x}\) | \(42\) |
trager | \(-\frac {\sqrt {1-2 x}}{5 \left (3+5 x \right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{275}\) | \(62\) |
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Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=\frac {\sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, \sqrt {-2 \, x + 1}}{275 \, {\left (5 \, x + 3\right )}} \]
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Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 175, normalized size of antiderivative = 3.65 \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=\begin {cases} \frac {2 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{275} + \frac {\sqrt {2}}{25 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {11 \sqrt {2}}{250 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {2 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{275} - \frac {\sqrt {2} i}{25 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {11 \sqrt {2} i}{250 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=-\frac {1}{275} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {\sqrt {-2 \, x + 1}}{5 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=-\frac {1}{275} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {\sqrt {-2 \, x + 1}}{5 \, {\left (5 \, x + 3\right )}} \]
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Time = 1.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx=\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{275}-\frac {2\,\sqrt {1-2\,x}}{25\,\left (2\,x+\frac {6}{5}\right )} \]
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