Optimal. Leaf size=81 \[ -\frac {39}{275} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{110 (3+5 x)^2}-\frac {13 (1-2 x)^{3/2}}{110 (3+5 x)}+\frac {39 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 52, 65,
212} \begin {gather*} -\frac {(1-2 x)^{5/2}}{110 (5 x+3)^2}-\frac {13 (1-2 x)^{3/2}}{110 (5 x+3)}-\frac {39}{275} \sqrt {1-2 x}+\frac {39 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{5/2}}{110 (3+5 x)^2}+\frac {13}{22} \int \frac {(1-2 x)^{3/2}}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{110 (3+5 x)^2}-\frac {13 (1-2 x)^{3/2}}{110 (3+5 x)}-\frac {39}{110} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=-\frac {39}{275} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{110 (3+5 x)^2}-\frac {13 (1-2 x)^{3/2}}{110 (3+5 x)}-\frac {39}{50} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {39}{275} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{110 (3+5 x)^2}-\frac {13 (1-2 x)^{3/2}}{110 (3+5 x)}+\frac {39}{50} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {39}{275} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{110 (3+5 x)^2}-\frac {13 (1-2 x)^{3/2}}{110 (3+5 x)}+\frac {39 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 58, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {1-2 x} \left (82+205 x+120 x^2\right )}{50 (3+5 x)^2}+\frac {39 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 57, normalized size = 0.70
method | result | size |
risch | \(\frac {240 x^{3}+290 x^{2}-41 x -82}{50 \left (3+5 x \right )^{2} \sqrt {1-2 x}}+\frac {39 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(51\) |
derivativedivides | \(-\frac {12 \sqrt {1-2 x}}{125}-\frac {4 \left (-\frac {61 \left (1-2 x \right )^{\frac {3}{2}}}{20}+\frac {693 \sqrt {1-2 x}}{100}\right )}{5 \left (-6-10 x \right )^{2}}+\frac {39 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(57\) |
default | \(-\frac {12 \sqrt {1-2 x}}{125}-\frac {4 \left (-\frac {61 \left (1-2 x \right )^{\frac {3}{2}}}{20}+\frac {693 \sqrt {1-2 x}}{100}\right )}{5 \left (-6-10 x \right )^{2}}+\frac {39 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(57\) |
trager | \(-\frac {\left (120 x^{2}+205 x +82\right ) \sqrt {1-2 x}}{50 \left (3+5 x \right )^{2}}+\frac {39 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{2750}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 83, normalized size = 1.02 \begin {gather*} -\frac {39}{2750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {12}{125} \, \sqrt {-2 \, x + 1} + \frac {305 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 693 \, \sqrt {-2 \, x + 1}}{125 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 75, normalized size = 0.93 \begin {gather*} \frac {39 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (120 \, x^{2} + 205 \, x + 82\right )} \sqrt {-2 \, x + 1}}{2750 \, {\left (25 \, x^{2} + 30 \, x + 9\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 = 0.59, size = 77, normalized size = 0.95 \begin {gather*} -\frac {39}{2750} \, \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 {12}{125} \, \sqrt {-2 \, x + 1} + \frac {305 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 693 \, \sqrt {-2 \, x + 1}}{500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 63, normalized size = 0.78 \begin {gather*} \frac {39\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1375}-\frac {12\,\sqrt {1-2\,x}}{125}-\frac {\frac {693\,\sqrt {1-2\,x}}{3125}-\frac {61\,{\left (1-2\,x\right )}^{3/2}}{625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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