Integrand size = 22, antiderivative size = 76 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=\frac {12}{25} \sqrt {1-2 x}+\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}-\frac {12}{25} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 52, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=-\frac {12}{25} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{5/2}}{55 (5 x+3)}+\frac {4}{55} (1-2 x)^{3/2}+\frac {12}{25} \sqrt {1-2 x} \]
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Rule 52
Rule 65
Rule 79
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{5/2}}{55 (3+5 x)}+\frac {6}{11} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx \\ & = \frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}+\frac {6}{5} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx \\ & = \frac {12}{25} \sqrt {1-2 x}+\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}+\frac {66}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {12}{25} \sqrt {1-2 x}+\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}-\frac {66}{25} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {12}{25} \sqrt {1-2 x}+\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}-\frac {12}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=\frac {1}{125} \left (\frac {5 \sqrt {1-2 x} \left (41+60 x-20 x^2\right )}{3+5 x}-12 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \]
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Time = 0.99 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {40 x^{3}-140 x^{2}-22 x +41}{25 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {12 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\) | \(51\) |
pseudoelliptic | \(\frac {-12 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-5 \sqrt {1-2 x}\, \left (20 x^{2}-60 x -41\right )}{375+625 x}\) | \(52\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{25}+\frac {62 \sqrt {1-2 x}}{125}+\frac {22 \sqrt {1-2 x}}{625 \left (-\frac {6}{5}-2 x \right )}-\frac {12 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\) | \(54\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{25}+\frac {62 \sqrt {1-2 x}}{125}+\frac {22 \sqrt {1-2 x}}{625 \left (-\frac {6}{5}-2 x \right )}-\frac {12 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\) | \(54\) |
trager | \(-\frac {\left (20 x^{2}-60 x -41\right ) \sqrt {1-2 x}}{25 \left (3+5 x \right )}-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{125}\) | \(72\) |
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Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=\frac {6 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (20 \, x^{2} - 60 \, x - 41\right )} \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
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Time = 28.74 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.43 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=\frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{25} + \frac {62 \sqrt {1 - 2 x}}{125} + \frac {29 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{625} - \frac {484 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{125} \]
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Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=\frac {2}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {62}{125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=\frac {2}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {62}{125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx=\frac {62\,\sqrt {1-2\,x}}{125}-\frac {22\,\sqrt {1-2\,x}}{625\,\left (2\,x+\frac {6}{5}\right )}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{25}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,12{}\mathrm {i}}{125} \]
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