Integrand size = 24, antiderivative size = 118 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=-\frac {103 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {207 \sqrt {1-2 x}}{2 (3+5 x)}+204 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {6933 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \]
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Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 156, 162, 65, 212} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=204 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {6933 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}}+\frac {207 \sqrt {1-2 x}}{2 (5 x+3)}-\frac {103 \sqrt {1-2 x}}{6 (5 x+3)^2}+\frac {7 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2} \]
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Rule 65
Rule 100
Rule 156
Rule 162
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {1}{3} \int \frac {124-171 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx \\ & = -\frac {103 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}-\frac {1}{66} \int \frac {8910-10197 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx \\ & = -\frac {103 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {207 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {1}{726} \int \frac {368082-225423 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx \\ & = -\frac {103 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {207 \sqrt {1-2 x}}{2 (3+5 x)}-2142 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {6933}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {103 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {207 \sqrt {1-2 x}}{2 (3+5 x)}+2142 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {6933}{2} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {103 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {207 \sqrt {1-2 x}}{2 (3+5 x)}+204 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {6933 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (1178+3830 x+3105 x^2\right )}{2 (2+3 x) (3+5 x)^2}+204 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {6933 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}} \]
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Time = 3.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {6210 x^{3}+4555 x^{2}-1474 x -1178}{2 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )}+204 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}-\frac {6933 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{55}\) | \(76\) |
derivativedivides | \(\frac {-685 \left (1-2 x \right )^{\frac {3}{2}}+1485 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {6933 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{55}-\frac {14 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}+204 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}\) | \(82\) |
default | \(\frac {-685 \left (1-2 x \right )^{\frac {3}{2}}+1485 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {6933 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{55}-\frac {14 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}+204 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}\) | \(82\) |
pseudoelliptic | \(\frac {22440 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {21}-13866 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {55}+55 \sqrt {1-2 x}\, \left (3105 x^{2}+3830 x +1178\right )}{110 \left (2+3 x \right ) \left (3+5 x \right )^{2}}\) | \(97\) |
trager | \(\frac {\left (3105 x^{2}+3830 x +1178\right ) \sqrt {1-2 x}}{2 \left (3+5 x \right )^{2} \left (2+3 x \right )}+\frac {6933 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{110}-102 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )\) | \(123\) |
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Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {6933 \, \sqrt {55} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11220 \, \sqrt {21} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 55 \, {\left (3105 \, x^{2} + 3830 \, x + 1178\right )} \sqrt {-2 \, x + 1}}{110 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
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Time = 63.41 (sec) , antiderivative size = 486, normalized size of antiderivative = 4.12 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=- 101 \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 ) + \frac {707 \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 )}{11} + 588 \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 ) + 3080 \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 ) + 968 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.08 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {6933}{110} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - 102 \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3105 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 13870 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 15477 \, \sqrt {-2 \, x + 1}}{75 \, {\left (2 \, x - 1\right )}^{3} + 505 \, {\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286} \]
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Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {6933}{110} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - 102 \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {21 \, \sqrt {-2 \, x + 1}}{3 \, x + 2} - \frac {5 \, {\left (137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 297 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} \]
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Time = 1.60 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=204\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )-\frac {6933\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{55}+\frac {\frac {5159\,\sqrt {1-2\,x}}{25}-\frac {2774\,{\left (1-2\,x\right )}^{3/2}}{15}+\frac {207\,{\left (1-2\,x\right )}^{5/2}}{5}}{\frac {2266\,x}{75}+\frac {101\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {286}{75}} \]
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