Optimal. Leaf size=131 \[ -\frac {6763 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 156,
162, 65, 212} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}+\frac {343 \sqrt {1-2 x}}{9 (3 x+2) (5 x+3)}-\frac {6763 \sqrt {1-2 x}}{18 (5 x+3)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 154
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {1}{6} \int \frac {(164-97 x) \sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}-\frac {1}{18} \int \frac {-8821+10096 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {6763 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}+\frac {1}{198} \int \frac {-364419+223179 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {6763 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}+\frac {46655}{6} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-12584 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {6763 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}-\frac {46655}{6} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+12584 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {6763 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 94, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {1-2 x} \left (8553+26380 x+20289 x^2\right )}{6 (2+3 x)^2 (3+5 x)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 82, normalized size = 0.63
method | result | size |
risch | \(\frac {40578 x^{3}+32471 x^{2}-9274 x -8553}{6 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}+\frac {2288 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}-\frac {6665 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\) | \(76\) |
derivativedivides | \(\frac {242 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {2288 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}+\frac {917 \left (1-2 x \right )^{\frac {3}{2}}-\frac {6517 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{2}}-\frac {6665 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\) | \(82\) |
default | \(\frac {242 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {2288 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}+\frac {917 \left (1-2 x \right )^{\frac {3}{2}}-\frac {6517 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{2}}-\frac {6665 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\) | \(82\) |
trager | \(-\frac {\left (20289 x^{2}+26380 x +8553\right ) \sqrt {1-2 x}}{6 \left (2+3 x \right )^{2} \left (3+5 x \right )}-\frac {6665 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{18}+\frac {1144 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{5}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 128, normalized size = 0.98 \begin {gather*} -\frac {1144}{5} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {6665}{18} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20289 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 93338 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 107261 \, \sqrt {-2 \, x + 1}}{3 \, {\left (45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.04, size = 142, normalized size = 1.08 \begin {gather*} \frac {20592 \, \sqrt {11} \sqrt {5} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 33325 \, \sqrt {7} \sqrt {3} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 15 \, {\left (20289 \, x^{2} + 26380 \, x + 8553\right )} \sqrt {-2 \, x + 1}}{90 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 123, normalized size = 0.94 \begin {gather*} -\frac {1144}{5} \, \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 {6665}{18} \, \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 {121 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} + \frac {7 \, {\left (393 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 931 \, \sqrt {-2 \, x + 1}\right )}}{12 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 90, normalized size = 0.69 \begin {gather*} \frac {2288\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{5}-\frac {6665\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9}-\frac {\frac {107261\,\sqrt {1-2\,x}}{135}-\frac {93338\,{\left (1-2\,x\right )}^{3/2}}{135}+\frac {6763\,{\left (1-2\,x\right )}^{5/2}}{45}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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