Optimal. Leaf size=83 \[ \frac {3}{25} \sqrt {1-2 x}-\frac {(1-2 x)^{5/2}}{10 (3+5 x)^2}+\frac {(1-2 x)^{3/2}}{10 (3+5 x)}-\frac {3}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.01, 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 = {43, 52, 65, 212}
\begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 212
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} \text {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 [A]
time = 0.14, size = 58, normalized size = 0.70 \begin {gather*} \frac {1}{250} \left (\frac {5 \sqrt {1-2 x} \left (64+195 x+80 x^2\right )}{(3+5 x)^2}-6 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 57, normalized size = 0.69
method | result | size |
risch | \(-\frac {160 x^{3}+310 x^{2}-67 x -64}{50 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {3 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\) | \(51\) |
derivativedivides | \(\frac {8 \sqrt {1-2 x}}{125}+\frac {-\frac {99 \left (1-2 x \right )^{\frac {3}{2}}}{25}+\frac {847 \sqrt {1-2 x}}{125}}{\left (-6-10 x \right )^{2}}-\frac {3 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\) | \(57\) |
default | \(\frac {8 \sqrt {1-2 x}}{125}+\frac {-\frac {99 \left (1-2 x \right )^{\frac {3}{2}}}{25}+\frac {847 \sqrt {1-2 x}}{125}}{\left (-6-10 x \right )^{2}}-\frac {3 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{125}\) | \(57\) |
trager | \(\frac {\left (80 x^{2}+195 x +64\right ) \sqrt {1-2 x}}{50 \left (3+5 x \right )^{2}}-\frac {3 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{250}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 83, normalized size = 1.00 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.84, size = 80, normalized size = 0.96 \begin {gather*} \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 )}} \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 = 3.08, size = 298, normalized size = 3.59 \begin {gather*} \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 {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 )}}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 77, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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