Optimal. Leaf size=76 \[ -\frac {(1-2 x)^{5/2}}{5 (5 x+3)}-\frac {2}{15} (1-2 x)^{3/2}-\frac {22}{25} \sqrt {1-2 x}+\frac {22}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {47, 50, 63, 206} \[ -\frac {(1-2 x)^{5/2}}{5 (5 x+3)}-\frac {2}{15} (1-2 x)^{3/2}-\frac {22}{25} \sqrt {1-2 x}+\frac {22}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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)^{5/2}}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{5/2}}{5 (3+5 x)}-\int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=-\frac {2}{15} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{5 (3+5 x)}-\frac {11}{5} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=-\frac {22}{25} \sqrt {1-2 x}-\frac {2}{15} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{5 (3+5 x)}-\frac {121}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {22}{25} \sqrt {1-2 x}-\frac {2}{15} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{5 (3+5 x)}+\frac {121}{25} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {22}{25} \sqrt {1-2 x}-\frac {2}{15} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{5 (3+5 x)}+\frac {22}{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 = 0.39 \[ -\frac {4}{847} (1-2 x)^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};-\frac {5}{11} (2 x-1)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 71, normalized size = 0.93 \[ \frac {33 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 5 \, {\left (40 \, x^{2} - 260 \, x - 243\right )} \sqrt {-2 \, x + 1}}{375 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 74, normalized size = 0.97 \[ -\frac {4}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {11}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {88}{125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.71 \[ \frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{125}-\frac {4 \left (-2 x +1\right )^{\frac {3}{2}}}{75}-\frac {88 \sqrt {-2 x +1}}{125}+\frac {242 \sqrt {-2 x +1}}{625 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 71, normalized size = 0.93 \[ -\frac {4}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {11}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {88}{125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 55, normalized size = 0.72 \[ -\frac {242\,\sqrt {1-2\,x}}{625\,\left (2\,x+\frac {6}{5}\right )}-\frac {88\,\sqrt {1-2\,x}}{125}-\frac {4\,{\left (1-2\,x\right )}^{3/2}}{75}-\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{125} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.51, size = 197, normalized size = 2.59 \[ \begin {cases} \frac {8 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{375} - \frac {308 \sqrt {5} i \sqrt {10 x - 5}}{1875} - \frac {22 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} - \frac {121 \sqrt {5} i \sqrt {10 x - 5}}{3125 \left (x + \frac {3}{5}\right )} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {8 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{375} - \frac {308 \sqrt {5} \sqrt {5 - 10 x}}{1875} - \frac {121 \sqrt {5} \sqrt {5 - 10 x}}{3125 \left (x + \frac {3}{5}\right )} - \frac {11 \sqrt {55} \log {\left (x + \frac {3}{5} \right )}}{125} + \frac {22 \sqrt {55} \log {\left (\sqrt {\frac {5}{11} - \frac {10 x}{11}} + 1 \right )}}{125} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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