Optimal. Leaf size=68 \[ -\frac {(1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{50 (3+5 x)}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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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 = {43, 65, 212}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 212
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} \text {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 = 0.15, size = 53, normalized size = 0.78 \begin {gather*} \frac {\sqrt {1-2 x} (4+25 x)}{50 (3+5 x)^2}-\frac {3 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 48, normalized size = 0.71
method | result | size |
risch | \(-\frac {50 x^{2}-17 x -4}{50 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {3 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(46\) |
derivativedivides | \(\frac {-\left (1-2 x \right )^{\frac {3}{2}}+\frac {33 \sqrt {1-2 x}}{25}}{\left (-6-10 x \right )^{2}}-\frac {3 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(48\) |
default | \(\frac {-\left (1-2 x \right )^{\frac {3}{2}}+\frac {33 \sqrt {1-2 x}}{25}}{\left (-6-10 x \right )^{2}}-\frac {3 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(48\) |
trager | \(\frac {\left (4+25 x \right ) \sqrt {1-2 x}}{50 \left (3+5 x \right )^{2}}-\frac {3 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{2750}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 74, normalized size = 1.09 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 69, normalized size = 1.01 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.34, size = 235, normalized size = 3.46 \begin {gather*} \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 {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 68, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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