Integrand size = 24, antiderivative size = 54 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {90, 65, 212} \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}}+\frac {3}{10} (1-2 x)^{3/2}-\frac {111}{50} \sqrt {1-2 x} \]
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Rule 65
Rule 90
Rule 212
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {111}{50 \sqrt {1-2 x}}-\frac {9}{10} \sqrt {1-2 x}+\frac {1}{25 \sqrt {1-2 x} (3+5 x)}\right ) \, dx \\ & = -\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}+\frac {1}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {1}{25} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {3}{25} \sqrt {1-2 x} (16+5 x)-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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Time = 1.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}+\frac {3 \left (-16-5 x \right ) \sqrt {1-2 x}}{25}\) | \(34\) |
derivativedivides | \(\frac {3 \left (1-2 x \right )^{\frac {3}{2}}}{10}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}-\frac {111 \sqrt {1-2 x}}{50}\) | \(38\) |
default | \(\frac {3 \left (1-2 x \right )^{\frac {3}{2}}}{10}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}-\frac {111 \sqrt {1-2 x}}{50}\) | \(38\) |
risch | \(\frac {3 \left (5 x +16\right ) \left (-1+2 x \right )}{25 \sqrt {1-2 x}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(39\) |
trager | \(\left (-\frac {3 x}{5}-\frac {48}{25}\right ) \sqrt {1-2 x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1375}\) | \(59\) |
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Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {3}{25} \, {\left (5 \, x + 16\right )} \sqrt {-2 \, x + 1} + \frac {1}{1375} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \]
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Time = 1.50 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {3 \left (1 - 2 x\right )^{\frac {3}{2}}}{10} - \frac {111 \sqrt {1 - 2 x}}{50} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{1375} \]
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Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {3}{10} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {111}{50} \, \sqrt {-2 \, x + 1} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {3}{10} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {111}{50} \, \sqrt {-2 \, x + 1} \]
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Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {3\,{\left (1-2\,x\right )}^{3/2}}{10}-\frac {111\,\sqrt {1-2\,x}}{50}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1375} \]
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