Integrand size = 17, antiderivative size = 56 \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=\frac {22}{25} \sqrt {1-2 x}+\frac {2}{15} (1-2 x)^{3/2}-\frac {22}{25} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 65, 212} \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=-\frac {22}{25} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+\frac {2}{15} (1-2 x)^{3/2}+\frac {22}{25} \sqrt {1-2 x} \]
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Rule 52
Rule 65
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {2}{15} (1-2 x)^{3/2}+\frac {11}{5} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx \\ & = \frac {22}{25} \sqrt {1-2 x}+\frac {2}{15} (1-2 x)^{3/2}+\frac {121}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {22}{25} \sqrt {1-2 x}+\frac {2}{15} (1-2 x)^{3/2}-\frac {121}{25} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {22}{25} \sqrt {1-2 x}+\frac {2}{15} (1-2 x)^{3/2}-\frac {22}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=\frac {1}{375} \left (20 (19-5 x) \sqrt {1-2 x}-66 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \]
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Time = 0.96 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}+\frac {4 \left (-5 x +19\right ) \sqrt {1-2 x}}{75}\) | \(34\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{15}-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}+\frac {22 \sqrt {1-2 x}}{25}\) | \(38\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{15}-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}+\frac {22 \sqrt {1-2 x}}{25}\) | \(38\) |
risch | \(\frac {4 \left (5 x -19\right ) \left (-1+2 x \right )}{75 \sqrt {1-2 x}}-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\) | \(39\) |
trager | \(\left (-\frac {4 x}{15}+\frac {76}{75}\right ) \sqrt {1-2 x}-\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{125}\) | \(59\) |
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=\frac {11}{125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {4}{75} \, {\left (5 \, x - 19\right )} \sqrt {-2 \, x + 1} \]
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Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.73 \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=\begin {cases} - \frac {4 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{75} + \frac {88 \sqrt {5} i \sqrt {10 x - 5}}{375} + \frac {22 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {4 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{75} + \frac {88 \sqrt {5} \sqrt {5 - 10 x}}{375} + \frac {11 \sqrt {55} \log {\left (x + \frac {3}{5} \right )}}{125} - \frac {22 \sqrt {55} \log {\left (\sqrt {\frac {5}{11} - \frac {10 x}{11}} + 1 \right )}}{125} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=\frac {2}{15} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{25} \, \sqrt {-2 \, x + 1} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=\frac {2}{15} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22}{25} \, \sqrt {-2 \, x + 1} \]
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Time = 1.51 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx=\frac {22\,\sqrt {1-2\,x}}{25}-\frac {22\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{125}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{15} \]
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