Integrand size = 22, antiderivative size = 41 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {7}{11 \sqrt {1-2 x}}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{11 \sqrt {55}} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {79, 65, 212} \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {7}{11 \sqrt {1-2 x}}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{11 \sqrt {55}} \]
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Rule 65
Rule 79
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {7}{11 \sqrt {1-2 x}}+\frac {1}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {7}{11 \sqrt {1-2 x}}-\frac {1}{11} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {7}{11 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{11 \sqrt {55}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {7}{11 \sqrt {1-2 x}}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{11 \sqrt {55}} \]
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Time = 1.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{605}+\frac {7}{11 \sqrt {1-2 x}}\) | \(29\) |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{605}+\frac {7}{11 \sqrt {1-2 x}}\) | \(29\) |
risch | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{605}+\frac {7}{11 \sqrt {1-2 x}}\) | \(29\) |
pseudoelliptic | \(-\frac {2 \left (\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\, \sqrt {1-2 x}-\frac {385}{2}\right )}{605 \sqrt {1-2 x}}\) | \(36\) |
trager | \(-\frac {7 \sqrt {1-2 x}}{11 \left (-1+2 x \right )}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{605}\) | \(62\) |
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Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {\sqrt {55} {\left (2 \, x - 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 385 \, \sqrt {-2 \, x + 1}}{605 \, {\left (2 \, x - 1\right )}} \]
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Time = 1.82 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{605} + \frac {7}{11 \sqrt {1 - 2 x}} \]
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Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {1}{605} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {7}{11 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.20 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {1}{605} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {7}{11 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {7}{11\,\sqrt {1-2\,x}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{605} \]
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