Optimal. Leaf size=68 \[ -\frac {3 \sqrt {1-2 x}}{242 (5 x+3)}-\frac {\sqrt {1-2 x}}{22 (5 x+3)^2}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
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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 = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {51, 63, 206} \[ -\frac {3 \sqrt {1-2 x}}{242 (5 x+3)}-\frac {\sqrt {1-2 x}}{22 (5 x+3)^2}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x}}{22 (3+5 x)^2}+\frac {3}{22} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x}}{22 (3+5 x)^2}-\frac {3 \sqrt {1-2 x}}{242 (3+5 x)}+\frac {3}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {\sqrt {1-2 x}}{22 (3+5 x)^2}-\frac {3 \sqrt {1-2 x}}{242 (3+5 x)}-\frac {3}{242} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {\sqrt {1-2 x}}{22 (3+5 x)^2}-\frac {3 \sqrt {1-2 x}}{242 (3+5 x)}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.44 \[ -\frac {8 \sqrt {1-2 x} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {5}{11} (2 x-1)\right )}{1331} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 69, normalized size = 1.01 \[ \frac {3 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 275 \, {\left (3 \, x + 4\right )} \sqrt {-2 \, x + 1}}{13310 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 68, normalized size = 1.00 \[ \frac {3}{13310} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {5 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}}{484 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.76 \[ -\frac {3 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{6655}-\frac {2 \sqrt {-2 x +1}}{11 \left (-10 x -6\right )^{2}}+\frac {3 \sqrt {-2 x +1}}{121 \left (-10 x -6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 74, normalized size = 1.09 \[ \frac {3}{13310} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {5 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}}{121 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 54, normalized size = 0.79 \[ -\frac {3\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{6655}-\frac {\frac {\sqrt {1-2\,x}}{55}-\frac {3\,{\left (1-2\,x\right )}^{3/2}}{605}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.06, size = 231, normalized size = 3.40 \[ \begin {cases} - \frac {3 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{6655} + \frac {3 \sqrt {2}}{1210 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {\sqrt {2}}{1100 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} - \frac {\sqrt {2}}{500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {3 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{6655} - \frac {3 \sqrt {2} i}{1210 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {\sqrt {2} i}{1100 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} + \frac {\sqrt {2} i}{500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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