Integrand size = 17, antiderivative size = 63 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (3+5 x)}-\frac {6}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {44, 53, 65, 212} \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=-\frac {6}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+\frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (5 x+3)} \]
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Rule 44
Rule 53
Rule 65
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{11 \sqrt {1-2 x} (3+5 x)}+\frac {3}{11} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx \\ & = \frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (3+5 x)}+\frac {15}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (3+5 x)}-\frac {15}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (3+5 x)}-\frac {6}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {2 (-22+15 (1-2 x))}{121 (-11+5 (1-2 x)) \sqrt {1-2 x}}-\frac {6}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 1.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {30 x +7}{121 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}\) | \(41\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {4}{121 \sqrt {1-2 x}}\) | \(45\) |
default | \(\frac {2 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {4}{121 \sqrt {1-2 x}}\) | \(45\) |
pseudoelliptic | \(-\frac {30 \left (\sqrt {55}\, \left (x +\frac {3}{5}\right ) \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-11 x -\frac {77}{30}\right )}{\sqrt {1-2 x}\, \left (3993+6655 x \right )}\) | \(49\) |
trager | \(-\frac {\left (30 x +7\right ) \sqrt {1-2 x}}{121 \left (10 x^{2}+x -3\right )}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(70\) |
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Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {3 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (30 \, x + 7\right )} \sqrt {-2 \, x + 1}}{1331 \, {\left (10 \, x^{2} + x - 3\right )}} \]
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Result contains complex when optimal does not.
Time = 1.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.78 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\begin {cases} - \frac {6 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1331} + \frac {3 \sqrt {2}}{121 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {\sqrt {2}}{110 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\frac {6 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1331} - \frac {3 \sqrt {2} i}{121 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {\sqrt {2} i}{110 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {3}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {3}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 1.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {\frac {12\,x}{121}+\frac {14}{605}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {6\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]
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