Integrand size = 24, antiderivative size = 67 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]
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Time = 0.02 (sec) , antiderivative size = 67, 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} \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}}-\frac {27}{100} (1-2 x)^{5/2}+\frac {54}{25} (1-2 x)^{3/2}-\frac {3897}{500} \sqrt {1-2 x} \]
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Rule 65
Rule 90
Rule 212
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3897}{500 \sqrt {1-2 x}}-\frac {162}{25} \sqrt {1-2 x}+\frac {27}{20} (1-2 x)^{3/2}+\frac {1}{125 \sqrt {1-2 x} (3+5 x)}\right ) \, dx \\ & = -\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}+\frac {1}{125} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}-\frac {1}{125} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {9}{125} \sqrt {1-2 x} \left (82+45 x+15 x^2\right )-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]
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Time = 1.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}-\frac {9 \sqrt {1-2 x}\, \left (15 x^{2}+45 x +82\right )}{125}\) | \(39\) |
risch | \(\frac {9 \left (15 x^{2}+45 x +82\right ) \left (-1+2 x \right )}{125 \sqrt {1-2 x}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\) | \(44\) |
derivativedivides | \(\frac {54 \left (1-2 x \right )^{\frac {3}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{100}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}-\frac {3897 \sqrt {1-2 x}}{500}\) | \(47\) |
default | \(\frac {54 \left (1-2 x \right )^{\frac {3}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{100}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}-\frac {3897 \sqrt {1-2 x}}{500}\) | \(47\) |
trager | \(\left (-\frac {27}{25} x^{2}-\frac {81}{25} x -\frac {738}{125}\right ) \sqrt {1-2 x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{6875}\) | \(64\) |
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Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {9}{125} \, {\left (15 \, x^{2} + 45 \, x + 82\right )} \sqrt {-2 \, x + 1} + \frac {1}{6875} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \]
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Time = 1.70 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=- \frac {27 \left (1 - 2 x\right )^{\frac {5}{2}}}{100} + \frac {54 \left (1 - 2 x\right )^{\frac {3}{2}}}{25} - \frac {3897 \sqrt {1 - 2 x}}{500} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{6875} \]
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Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {27}{100} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {54}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{6875} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3897}{500} \, \sqrt {-2 \, x + 1} \]
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Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {27}{100} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {54}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{6875} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3897}{500} \, \sqrt {-2 \, x + 1} \]
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Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {54\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {3897\,\sqrt {1-2\,x}}{500}-\frac {27\,{\left (1-2\,x\right )}^{5/2}}{100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{6875} \]
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