Optimal. Leaf size=63 \[ -\frac {(1-2 x)^{3/2}}{5 (5 x+3)}-\frac {6}{25} \sqrt {1-2 x}+\frac {6}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {47, 50, 63, 206} \[ -\frac {(1-2 x)^{3/2}}{5 (5 x+3)}-\frac {6}{25} \sqrt {1-2 x}+\frac {6}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}-\frac {3}{5} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=-\frac {6}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}-\frac {33}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {6}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}+\frac {33}{25} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {6}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{5 (3+5 x)}+\frac {6}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.48 \[ -\frac {4}{605} (1-2 x)^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};-\frac {5}{11} (2 x-1)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 66, normalized size = 1.05 \[ \frac {3 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) - 5 \, {\left (20 \, x + 23\right )} \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 65, normalized size = 1.03 \[ -\frac {3}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {4}{25} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.71 \[ \frac {6 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{125}-\frac {4 \sqrt {-2 x +1}}{25}+\frac {22 \sqrt {-2 x +1}}{125 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 62, normalized size = 0.98 \[ -\frac {3}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4}{25} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 44, normalized size = 0.70 \[ \frac {6\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{125}-\frac {22\,\sqrt {1-2\,x}}{125\,\left (2\,x+\frac {6}{5}\right )}-\frac {4\,\sqrt {1-2\,x}}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.90, size = 238, normalized size = 3.78 \[ \begin {cases} \frac {6 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} + \frac {4 \sqrt {2} \sqrt {x + \frac {3}{5}}}{25 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}} - \frac {11 \sqrt {2}}{125 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {121 \sqrt {2}}{1250 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\- \frac {6 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} - \frac {4 \sqrt {2} i \sqrt {x + \frac {3}{5}}}{25 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}} + \frac {11 \sqrt {2} i}{125 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {121 \sqrt {2} i}{1250 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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