Integrand size = 22, antiderivative size = 108 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \]
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Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 44, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=-\frac {137 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}}+\frac {(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac {137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac {137 \sqrt {1-2 x}}{10584 (3 x+2)}+\frac {137 \sqrt {1-2 x}}{1512 (3 x+2)^2} \]
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Rule 43
Rule 44
Rule 65
Rule 79
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}+\frac {137}{84} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}-\frac {137}{252} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1512} \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{10584} \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{10584} \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (970-7990 x-13245 x^2+3699 x^3\right )}{2 (2+3 x)^4}-137 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132} \]
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Time = 0.98 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {7398 x^{4}-30189 x^{3}-2735 x^{2}+9930 x -970}{10584 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {137 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{111132}\) | \(56\) |
pseudoelliptic | \(\frac {-274 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (3699 x^{3}-13245 x^{2}-7990 x +970\right )}{222264 \left (2+3 x \right )^{4}}\) | \(60\) |
derivativedivides | \(-\frac {1296 \left (-\frac {137 \left (1-2 x \right )^{\frac {7}{2}}}{254016}-\frac {733 \left (1-2 x \right )^{\frac {5}{2}}}{326592}+\frac {1507 \left (1-2 x \right )^{\frac {3}{2}}}{139968}-\frac {959 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{4}}-\frac {137 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{111132}\) | \(66\) |
default | \(-\frac {1296 \left (-\frac {137 \left (1-2 x \right )^{\frac {7}{2}}}{254016}-\frac {733 \left (1-2 x \right )^{\frac {5}{2}}}{326592}+\frac {1507 \left (1-2 x \right )^{\frac {3}{2}}}{139968}-\frac {959 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{4}}-\frac {137 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{111132}\) | \(66\) |
trager | \(-\frac {\left (3699 x^{3}-13245 x^{2}-7990 x +970\right ) \sqrt {1-2 x}}{10584 \left (2+3 x \right )^{4}}+\frac {137 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{222264}\) | \(77\) |
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Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {137 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (3699 \, x^{3} - 13245 \, x^{2} - 7990 \, x + 970\right )} \sqrt {-2 \, x + 1}}{222264 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {137}{222264} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3699 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 15393 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 46991 \, \sqrt {-2 \, x + 1}}{5292 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {137}{222264} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3699 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 15393 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 46991 \, \sqrt {-2 \, x + 1}}{84672 \, {\left (3 \, x + 2\right )}^{4}} \]
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Time = 1.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {\frac {959\,\sqrt {1-2\,x}}{8748}-\frac {1507\,{\left (1-2\,x\right )}^{3/2}}{8748}+\frac {733\,{\left (1-2\,x\right )}^{5/2}}{20412}+\frac {137\,{\left (1-2\,x\right )}^{7/2}}{15876}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}}-\frac {137\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{111132} \]
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