Integrand size = 17, antiderivative size = 69 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {242}{125} \sqrt {1-2 x}+\frac {22}{75} (1-2 x)^{3/2}+\frac {2}{25} (1-2 x)^{5/2}-\frac {242}{125} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 65, 212} \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=-\frac {242}{125} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+\frac {2}{25} (1-2 x)^{5/2}+\frac {22}{75} (1-2 x)^{3/2}+\frac {242}{125} \sqrt {1-2 x} \]
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Rule 52
Rule 65
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {2}{25} (1-2 x)^{5/2}+\frac {11}{5} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx \\ & = \frac {22}{75} (1-2 x)^{3/2}+\frac {2}{25} (1-2 x)^{5/2}+\frac {121}{25} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx \\ & = \frac {242}{125} \sqrt {1-2 x}+\frac {22}{75} (1-2 x)^{3/2}+\frac {2}{25} (1-2 x)^{5/2}+\frac {1331}{125} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {242}{125} \sqrt {1-2 x}+\frac {22}{75} (1-2 x)^{3/2}+\frac {2}{25} (1-2 x)^{5/2}-\frac {1331}{125} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {242}{125} \sqrt {1-2 x}+\frac {22}{75} (1-2 x)^{3/2}+\frac {2}{25} (1-2 x)^{5/2}-\frac {242}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (433-170 x+60 x^2\right )-726 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1875} \]
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Time = 1.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
pseudoelliptic | \(-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}+\frac {2 \sqrt {1-2 x}\, \left (60 x^{2}-170 x +433\right )}{375}\) | \(39\) |
risch | \(-\frac {2 \left (60 x^{2}-170 x +433\right ) \left (-1+2 x \right )}{375 \sqrt {1-2 x}}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}\) | \(44\) |
derivativedivides | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{75}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{25}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}+\frac {242 \sqrt {1-2 x}}{125}\) | \(47\) |
default | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{75}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{25}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}+\frac {242 \sqrt {1-2 x}}{125}\) | \(47\) |
trager | \(\left (\frac {8}{25} x^{2}-\frac {68}{75} x +\frac {866}{375}\right ) \sqrt {1-2 x}-\frac {121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{625}\) | \(64\) |
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Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {121}{625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {2}{375} \, {\left (60 \, x^{2} - 170 \, x + 433\right )} \sqrt {-2 \, x + 1} \]
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Result contains complex when optimal does not.
Time = 2.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.93 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\begin {cases} \frac {8 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{125} - \frac {484 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{1875} + \frac {5566 \sqrt {5} i \sqrt {10 x - 5}}{9375} + \frac {242 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{625} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {8 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{125} - \frac {484 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{1875} + \frac {5566 \sqrt {5} \sqrt {5 - 10 x}}{9375} + \frac {121 \sqrt {55} \log {\left (x + \frac {3}{5} \right )}}{625} - \frac {242 \sqrt {55} \log {\left (\sqrt {\frac {5}{11} - \frac {10 x}{11}} + 1 \right )}}{625} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {2}{25} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{125} \, \sqrt {-2 \, x + 1} \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {2}{25} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {242}{125} \, \sqrt {-2 \, x + 1} \]
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Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {242\,\sqrt {1-2\,x}}{125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{75}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{25}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{625} \]
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