Integrand size = 24, antiderivative size = 102 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {1342 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
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Time = 0.02 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 81, 52, 65, 212} \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=-\frac {1342 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}-\frac {9}{175} (1-2 x)^{7/2}+\frac {122 (1-2 x)^{5/2}}{6875}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {1342 \sqrt {1-2 x}}{3125} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {1}{275} \int \frac {(1-2 x)^{5/2} (358+495 x)}{3+5 x} \, dx \\ & = -\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {61}{275} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx \\ & = \frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {61}{125} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx \\ & = \frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {671}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx \\ & = \frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {7381 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125} \\ & = \frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {7381 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125} \\ & = \frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {1342 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (90486+173795 x-75130 x^2-96300 x^3+135000 x^4\right )}{3+5 x}-28182 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \]
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Time = 3.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.60
method | result | size |
risch | \(-\frac {270000 x^{5}-327600 x^{4}-53960 x^{3}+422720 x^{2}+7177 x -90486}{65625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {1342 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(61\) |
pseudoelliptic | \(\frac {-28182 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}+5 \sqrt {1-2 x}\, \left (135000 x^{4}-96300 x^{3}-75130 x^{2}+173795 x +90486\right )}{984375+1640625 x}\) | \(62\) |
derivativedivides | \(-\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{175}+\frac {12 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {128 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {1364 \sqrt {1-2 x}}{3125}+\frac {242 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {1342 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(72\) |
default | \(-\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{175}+\frac {12 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {128 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {1364 \sqrt {1-2 x}}{3125}+\frac {242 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {1342 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(72\) |
trager | \(\frac {\sqrt {1-2 x}\, \left (135000 x^{4}-96300 x^{3}-75130 x^{2}+173795 x +90486\right )}{196875+328125 x}+\frac {671 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{15625}\) | \(82\) |
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Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {14091 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (135000 \, x^{4} - 96300 \, x^{3} - 75130 \, x^{2} + 173795 \, x + 90486\right )} \sqrt {-2 \, x + 1}}{328125 \, {\left (5 \, x + 3\right )}} \]
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Time = 30.69 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.05 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=- \frac {9 \left (1 - 2 x\right )^{\frac {7}{2}}}{175} + \frac {12 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} + \frac {128 \left (1 - 2 x\right )^{\frac {3}{2}}}{1875} + \frac {1364 \sqrt {1 - 2 x}}{3125} + \frac {132 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{3125} - \frac {5324 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{3125} \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=-\frac {9}{175} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {12}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {9}{175} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {12}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \]
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Time = 1.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {1364\,\sqrt {1-2\,x}}{3125}-\frac {242\,\sqrt {1-2\,x}}{15625\,\left (2\,x+\frac {6}{5}\right )}+\frac {128\,{\left (1-2\,x\right )}^{3/2}}{1875}+\frac {12\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {9\,{\left (1-2\,x\right )}^{7/2}}{175}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,1342{}\mathrm {i}}{15625} \]
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