Integrand size = 22, antiderivative size = 96 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac {365}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 96, 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, 52, 65, 212} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=\frac {365}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac {73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac {365}{567} (1-2 x)^{3/2}-\frac {365}{81} \sqrt {1-2 x} \]
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}+\frac {73}{42} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2} \, dx \\ & = \frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac {365}{126} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx \\ & = -\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac {365}{54} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx \\ & = -\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac {2555}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = -\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac {2555}{162} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac {365}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=\frac {1}{486} \left (\frac {3 \sqrt {1-2 x} \left (-3521-8731 x-4584 x^2+720 x^3\right )}{(2+3 x)^2}+730 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \]
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Time = 1.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.58
method | result | size |
risch | \(-\frac {1440 x^{4}-9888 x^{3}-12878 x^{2}+1689 x +3521}{162 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {365 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(56\) |
pseudoelliptic | \(\frac {730 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+3 \sqrt {1-2 x}\, \left (720 x^{3}-4584 x^{2}-8731 x -3521\right )}{486 \left (2+3 x \right )^{2}}\) | \(60\) |
derivativedivides | \(-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {32 \sqrt {1-2 x}}{9}-\frac {28 \left (-\frac {79 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {539 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {365 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(66\) |
default | \(-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {32 \sqrt {1-2 x}}{9}-\frac {28 \left (-\frac {79 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {539 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {365 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(66\) |
trager | \(\frac {\left (720 x^{3}-4584 x^{2}-8731 x -3521\right ) \sqrt {1-2 x}}{162 \left (2+3 x \right )^{2}}-\frac {365 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{486}\) | \(77\) |
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Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=\frac {365 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 3 \, {\left (720 \, x^{3} - 4584 \, x^{2} - 8731 \, x - 3521\right )} \sqrt {-2 \, x + 1}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
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Time = 77.77 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.70 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=- \frac {20 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {32 \sqrt {1 - 2 x}}{9} - \frac {74 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{81} - \frac {8036 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} - \frac {2744 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} \]
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Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {365}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {32}{9} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (237 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \sqrt {-2 \, x + 1}\right )}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {365}{486} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {32}{9} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (237 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \sqrt {-2 \, x + 1}\right )}}{324 \, {\left (3 \, x + 2\right )}^{2}} \]
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Time = 1.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {32\,\sqrt {1-2\,x}}{9}-\frac {20\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {\frac {3773\,\sqrt {1-2\,x}}{729}-\frac {553\,{\left (1-2\,x\right )}^{3/2}}{243}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,365{}\mathrm {i}}{243} \]
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