Optimal. Leaf size=68 \[ -\frac {(1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac {3 \sqrt {1-2 x}}{50 (5 x+3)}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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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 = {47, 63, 206} \[ -\frac {(1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac {3 \sqrt {1-2 x}}{50 (5 x+3)}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{3/2}}{10 (3+5 x)^2}-\frac {3}{10} \int \frac {\sqrt {1-2 x}}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{50 (3+5 x)}+\frac {3}{50} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{50 (3+5 x)}-\frac {3}{50} \operatorname {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}}{10 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{50 (3+5 x)}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 1.09 \[ \frac {55 \left (-50 x^2+17 x+4\right )+6 \sqrt {55} \sqrt {2 x-1} (5 x+3)^2 \tan ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{2750 \sqrt {1-2 x} (5 x+3)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 69, normalized size = 1.01 \[ \frac {3 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (25 \, x + 4\right )} \sqrt {-2 \, x + 1}}{2750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 68, normalized size = 1.00 \[ \frac {3}{2750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {25 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 33 \, \sqrt {-2 \, x + 1}}{100 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.71 \[ -\frac {3 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1375}+\frac {-\left (-2 x +1\right )^{\frac {3}{2}}+\frac {33 \sqrt {-2 x +1}}{25}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 74, normalized size = 1.09 \[ \frac {3}{2750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {25 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 33 \, \sqrt {-2 \, x + 1}}{25 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 53, normalized size = 0.78 \[ \frac {\frac {33\,\sqrt {1-2\,x}}{625}-\frac {{\left (1-2\,x\right )}^{3/2}}{25}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}-\frac {3\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1375} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.34, size = 235, normalized size = 3.46 \[ \begin {cases} - \frac {3 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1375} - \frac {\sqrt {2}}{50 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {77 \sqrt {2}}{2500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} - \frac {121 \sqrt {2}}{12500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {3 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1375} + \frac {\sqrt {2} i}{50 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {77 \sqrt {2} i}{2500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} + \frac {121 \sqrt {2} i}{12500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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