Optimal. Leaf size=68 \[ -\frac {(1-2 x)^{3/2}}{110 (3+5 x)^2}-\frac {67 \sqrt {1-2 x}}{550 (3+5 x)}+\frac {67 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 43, 65, 212}
\begin {gather*} -\frac {(1-2 x)^{3/2}}{110 (5 x+3)^2}-\frac {67 \sqrt {1-2 x}}{550 (5 x+3)}+\frac {67 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{3/2}}{110 (3+5 x)^2}+\frac {67}{110} \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{110 (3+5 x)^2}-\frac {67 \sqrt {1-2 x}}{550 (3+5 x)}-\frac {67}{550} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{110 (3+5 x)^2}-\frac {67 \sqrt {1-2 x}}{550 (3+5 x)}+\frac {67}{550} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {(1-2 x)^{3/2}}{110 (3+5 x)^2}-\frac {67 \sqrt {1-2 x}}{550 (3+5 x)}+\frac {67 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 53, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-2 x} (206+325 x)}{550 (3+5 x)^2}+\frac {67 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 48, normalized size = 0.71
method | result | size |
risch | \(\frac {650 x^{2}+87 x -206}{550 \left (3+5 x \right )^{2} \sqrt {1-2 x}}+\frac {67 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\) | \(46\) |
derivativedivides | \(-\frac {100 \left (-\frac {13 \left (1-2 x \right )^{\frac {3}{2}}}{1100}+\frac {67 \sqrt {1-2 x}}{2500}\right )}{\left (-6-10 x \right )^{2}}+\frac {67 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\) | \(48\) |
default | \(-\frac {100 \left (-\frac {13 \left (1-2 x \right )^{\frac {3}{2}}}{1100}+\frac {67 \sqrt {1-2 x}}{2500}\right )}{\left (-6-10 x \right )^{2}}+\frac {67 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\) | \(48\) |
trager | \(-\frac {\left (325 x +206\right ) \sqrt {1-2 x}}{550 \left (3+5 x \right )^{2}}-\frac {67 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{30250}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 74, normalized size = 1.09 \begin {gather*} -\frac {67}{30250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {325 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 737 \, \sqrt {-2 \, x + 1}}{275 \, {\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 = 1.15, size = 70, normalized size = 1.03 \begin {gather*} \frac {67 \, \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 (325 \, x + 206\right )} \sqrt {-2 \, x + 1}}{30250 \, {\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (56) = 112\).
time = 173.79, size = 348, normalized size = 5.12 \begin {gather*} - \frac {124 \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 )}{25} + \frac {88 \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 )}{25} - \frac {12 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.28, size = 68, normalized size = 1.00 \begin {gather*} -\frac {67}{30250} \, \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 {325 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 737 \, \sqrt {-2 \, x + 1}}{1100 \, {\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 = 1.18, size = 54, normalized size = 0.79 \begin {gather*} \frac {67\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{15125}-\frac {\frac {67\,\sqrt {1-2\,x}}{625}-\frac {13\,{\left (1-2\,x\right )}^{3/2}}{275}}{\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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