Optimal. Leaf size=92 \[ \frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {18}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {103, 152, 156, 63, 206} \[ \frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {18}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 152
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx &=-\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} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {750}{121} \operatorname {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*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.79 \[ \frac {2178 (5 x+3) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-35 \left (60 (5 x+3) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+11\right )}{847 \sqrt {1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 116, normalized size = 1.26 \[ \frac {7350 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 11979 \, \sqrt {7} \sqrt {3} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (390 \, x - 151\right )} \sqrt {-2 \, x + 1}}{65219 \, {\left (10 \, x^{2} + x - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.33, size = 107, normalized size = 1.16 \[ -\frac {150}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.68 \[ -\frac {18 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{49}+\frac {300 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {8}{847 \sqrt {-2 x +1}}+\frac {10 \sqrt {-2 x +1}}{121 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 101, normalized size = 1.10 \[ -\frac {150}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 64, normalized size = 0.70 \[ \frac {\frac {156\,x}{847}-\frac {302}{4235}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}+\frac {300\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.11, size = 376, normalized size = 4.09 \[ \frac {30030 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {3388 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {147000 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {239580 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {119790 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {73500 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {161700 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {263538 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {131769 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {80850 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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