Optimal. Leaf size=83 \[ -\frac {(1-2 x)^{5/2}}{10 (5 x+3)^2}+\frac {(1-2 x)^{3/2}}{10 (5 x+3)}+\frac {3}{25} \sqrt {1-2 x}-\frac {3}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 83, 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} \[ -\frac {(1-2 x)^{5/2}}{10 (5 x+3)^2}+\frac {(1-2 x)^{3/2}}{10 (5 x+3)}+\frac {3}{25} \sqrt {1-2 x}-\frac {3}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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)^3} \, dx &=-\frac {(1-2 x)^{5/2}}{10 (3+5 x)^2}-\frac {1}{2} \int \frac {(1-2 x)^{3/2}}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{10 (3+5 x)^2}+\frac {(1-2 x)^{3/2}}{10 (3+5 x)}+\frac {3}{10} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {3}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{10 (3+5 x)^2}+\frac {(1-2 x)^{3/2}}{10 (3+5 x)}+\frac {33}{50} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {3}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{10 (3+5 x)^2}+\frac {(1-2 x)^{3/2}}{10 (3+5 x)}-\frac {33}{50} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {3}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{10 (3+5 x)^2}+\frac {(1-2 x)^{3/2}}{10 (3+5 x)}-\frac {3}{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 = 0.36 \[ -\frac {8 (1-2 x)^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};-\frac {5}{11} (2 x-1)\right )}{9317} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 80, normalized size = 0.96 \[ \frac {3 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (80 \, x^{2} + 195 \, x + 64\right )} \sqrt {-2 \, x + 1}}{250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 77, normalized size = 0.93 \[ \frac {3}{250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {8}{125} \, \sqrt {-2 \, x + 1} - \frac {11 \, {\left (45 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 77 \, \sqrt {-2 \, x + 1}\right )}}{500 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.69 \[ -\frac {3 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{125}+\frac {8 \sqrt {-2 x +1}}{125}+\frac {-\frac {99 \left (-2 x +1\right )^{\frac {3}{2}}}{25}+\frac {847 \sqrt {-2 x +1}}{125}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 83, normalized size = 1.00 \[ \frac {3}{250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8}{125} \, \sqrt {-2 \, x + 1} - \frac {11 \, {\left (45 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 77 \, \sqrt {-2 \, x + 1}\right )}}{125 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 62, normalized size = 0.75 \[ \frac {8\,\sqrt {1-2\,x}}{125}-\frac {3\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{125}+\frac {\frac {847\,\sqrt {1-2\,x}}{3125}-\frac {99\,{\left (1-2\,x\right )}^{3/2}}{625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.97, size = 298, normalized size = 3.59 \[ \begin {cases} - \frac {3 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} - \frac {8 \sqrt {2} \sqrt {x + \frac {3}{5}}}{125 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}} - \frac {11 \sqrt {2}}{1250 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {1331 \sqrt {2}}{12500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} - \frac {1331 \sqrt {2}}{62500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {3 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{125} + \frac {8 \sqrt {2} i \sqrt {x + \frac {3}{5}}}{125 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}} + \frac {11 \sqrt {2} i}{1250 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {1331 \sqrt {2} i}{12500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} + \frac {1331 \sqrt {2} i}{62500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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