Integrand size = 24, antiderivative size = 107 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac {328715 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}} \]
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Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {99, 154, 150, 65, 212} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {328715 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}}-\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{378 (3 x+2)^3}-\frac {5 \sqrt {1-2 x} (110981 x+70429)}{222264 (3 x+2)^2} \]
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Rule 65
Rule 99
Rule 150
Rule 154
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^4} \, dx \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}+\frac {1}{756} \int \frac {(445-3145 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}-\frac {328715 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{222264} \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac {328715 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{222264} \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac {328715 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (2469626+11657098 x+18358575 x^2+9646695 x^3\right )}{2 (2+3 x)^4}+328715 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2333772} \]
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Time = 0.99 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {19293390 x^{4}+27070455 x^{3}+4955621 x^{2}-6717846 x -2469626}{222264 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {328715 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(56\) |
pseudoelliptic | \(\frac {657430 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (9646695 x^{3}+18358575 x^{2}+11657098 x +2469626\right )}{4667544 \left (2+3 x \right )^{4}}\) | \(60\) |
derivativedivides | \(-\frac {324 \left (-\frac {119095 \left (1-2 x \right )^{\frac {7}{2}}}{444528}+\frac {3126535 \left (1-2 x \right )^{\frac {5}{2}}}{1714608}-\frac {3040873 \left (1-2 x \right )^{\frac {3}{2}}}{734832}+\frac {328715 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{4}}+\frac {328715 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(66\) |
default | \(-\frac {324 \left (-\frac {119095 \left (1-2 x \right )^{\frac {7}{2}}}{444528}+\frac {3126535 \left (1-2 x \right )^{\frac {5}{2}}}{1714608}-\frac {3040873 \left (1-2 x \right )^{\frac {3}{2}}}{734832}+\frac {328715 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{4}}+\frac {328715 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(66\) |
trager | \(-\frac {\left (9646695 x^{3}+18358575 x^{2}+11657098 x +2469626\right ) \sqrt {1-2 x}}{222264 \left (2+3 x \right )^{4}}-\frac {328715 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4667544}\) | \(77\) |
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Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {328715 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (9646695 \, x^{3} + 18358575 \, x^{2} + 11657098 \, x + 2469626\right )} \sqrt {-2 \, x + 1}}{4667544 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {328715}{4667544} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9646695 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 65657235 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 149002777 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 112749245 \, \sqrt {-2 \, x + 1}}{111132 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {328715}{4667544} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {9646695 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 65657235 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 149002777 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 112749245 \, \sqrt {-2 \, x + 1}}{1778112 \, {\left (3 \, x + 2\right )}^{4}} \]
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Time = 1.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {328715\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2333772}-\frac {\frac {328715\,\sqrt {1-2\,x}}{26244}-\frac {3040873\,{\left (1-2\,x\right )}^{3/2}}{183708}+\frac {3126535\,{\left (1-2\,x\right )}^{5/2}}{428652}-\frac {119095\,{\left (1-2\,x\right )}^{7/2}}{111132}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
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