Optimal. Leaf size=69 \[ \frac {22}{125} \sqrt {1-2 x}+\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}-\frac {22}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {81, 52, 65, 212}
\begin {gather*} -\frac {3}{25} (1-2 x)^{5/2}+\frac {2}{75} (1-2 x)^{3/2}+\frac {22}{125} \sqrt {1-2 x}-\frac {22}{125} \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 52
Rule 65
Rule 81
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)}{3+5 x} \, dx &=-\frac {3}{25} (1-2 x)^{5/2}+\frac {1}{5} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}+\frac {11}{25} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {22}{125} \sqrt {1-2 x}+\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}+\frac {121}{125} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {22}{125} \sqrt {1-2 x}+\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}-\frac {121}{125} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {22}{125} \sqrt {1-2 x}+\frac {2}{75} (1-2 x)^{3/2}-\frac {3}{25} (1-2 x)^{5/2}-\frac {22}{125} \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 = 0.06, size = 51, normalized size = 0.74 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (31+160 x-180 x^2\right )-66 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 47, normalized size = 0.68
method | result | size |
risch | \(\frac {\left (180 x^{2}-160 x -31\right ) \left (-1+2 x \right )}{375 \sqrt {1-2 x}}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}\) | \(44\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{75}-\frac {3 \left (1-2 x \right )^{\frac {5}{2}}}{25}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}+\frac {22 \sqrt {1-2 x}}{125}\) | \(47\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{75}-\frac {3 \left (1-2 x \right )^{\frac {5}{2}}}{25}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}+\frac {22 \sqrt {1-2 x}}{125}\) | \(47\) |
trager | \(\left (-\frac {12}{25} x^{2}+\frac {32}{75} x +\frac {31}{375}\right ) \sqrt {1-2 x}-\frac {11 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{625}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 64, normalized size = 0.93 \begin {gather*} -\frac {3}{25} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 56, normalized size = 0.81 \begin {gather*} \frac {11}{625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{375} \, {\left (180 \, x^{2} - 160 \, x - 31\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.95, size = 95, normalized size = 1.38 \begin {gather*} - \frac {3 \left (1 - 2 x\right )^{\frac {5}{2}}}{25} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{75} + \frac {22 \sqrt {1 - 2 x}}{125} + \frac {242 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 74, normalized size = 1.07 \begin {gather*} -\frac {3}{25} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{625} \, \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 {22}{125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 48, normalized size = 0.70 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{125}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{75}-\frac {3\,{\left (1-2\,x\right )}^{5/2}}{25}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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