Integrand size = 24, antiderivative size = 92 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 92, 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 = {105, 157, 162, 65, 212} \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {18}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (5 x+3)} \]
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Rule 65
Rule 105
Rule 157
Rule 162
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{11} \int \frac {3-45 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx \\ & = \frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {2}{847} \int \frac {-\frac {699}{2}+\frac {585 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx \\ & = \frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {27}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {750}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {27}{7} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {750}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {-151+390 x}{847 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 1.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.64
method | result | size |
risch | \(\frac {390 x -151}{847 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}+\frac {300 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}\) | \(59\) |
derivativedivides | \(\frac {10 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {300 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}+\frac {8}{847 \sqrt {1-2 x}}\) | \(63\) |
default | \(\frac {10 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {300 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}+\frac {8}{847 \sqrt {1-2 x}}\) | \(63\) |
pseudoelliptic | \(-\frac {119790 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right ) \sqrt {21}}{5}-\frac {490 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}}{3993}-\frac {91 x}{363}+\frac {1057}{10890}\right )}{\sqrt {1-2 x}\, \left (195657+326095 x \right )}\) | \(82\) |
trager | \(-\frac {\left (390 x -151\right ) \sqrt {1-2 x}}{847 \left (10 x^{2}+x -3\right )}-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49}-\frac {150 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(114\) |
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Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {7350 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 11979 \, \sqrt {7} \sqrt {3} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (390 \, x - 151\right )} \sqrt {-2 \, x + 1}}{65219 \, {\left (10 \, x^{2} + x - 3\right )}} \]
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Result contains complex when optimal does not.
Time = 5.92 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=- \frac {30030 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {3388 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {147000 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {239580 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {73500 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {119790 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {161700 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {263538 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {80850 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {131769 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {150}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {150}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {\frac {156\,x}{847}-\frac {302}{4235}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}+\frac {300\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]
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