Optimal. Leaf size=95 \[ \frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}-\frac {2 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
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Rubi [A]
time = 0.02, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 212}
\begin {gather*} \frac {9}{40} (1-2 x)^{9/2}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {2 \sqrt {1-2 x}}{3125}-\frac {2 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx &=\int \left (\frac {136419 \sqrt {1-2 x}}{5000}-\frac {34371 (1-2 x)^{3/2}}{1000}+\frac {2889}{200} (1-2 x)^{5/2}-\frac {81}{40} (1-2 x)^{7/2}+\frac {\sqrt {1-2 x}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}+\frac {1}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}+\frac {11 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}-\frac {11 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}-\frac {2 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 61, normalized size = 0.64 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (-88776+27865 x+177930 x^2+203625 x^3+78750 x^4\right )-14 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{109375} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 65, normalized size = 0.68
method | result | size |
risch | \(-\frac {\left (78750 x^{4}+203625 x^{3}+177930 x^{2}+27865 x -88776\right ) \left (-1+2 x \right )}{21875 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(54\) |
derivativedivides | \(-\frac {45473 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {34371 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {2889 \left (1-2 x \right )^{\frac {7}{2}}}{1400}+\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{40}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {2 \sqrt {1-2 x}}{3125}\) | \(65\) |
default | \(-\frac {45473 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {34371 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {2889 \left (1-2 x \right )^{\frac {7}{2}}}{1400}+\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{40}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {2 \sqrt {1-2 x}}{3125}\) | \(65\) |
trager | \(\left (\frac {18}{5} x^{4}+\frac {1629}{175} x^{3}+\frac {35586}{4375} x^{2}+\frac {5573}{4375} x -\frac {88776}{21875}\right ) \sqrt {1-2 x}+\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{15625}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 82, normalized size = 0.86 \begin {gather*} \frac {9}{40} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {2889}{1400} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {34371}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {45473}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.94, size = 66, normalized size = 0.69 \begin {gather*} \frac {1}{15625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{21875} \, {\left (78750 \, x^{4} + 203625 \, x^{3} + 177930 \, x^{2} + 27865 \, x - 88776\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.91, size = 119, normalized size = 1.25 \begin {gather*} \frac {9 \left (1 - 2 x\right )^{\frac {9}{2}}}{40} - \frac {2889 \left (1 - 2 x\right )^{\frac {7}{2}}}{1400} + \frac {34371 \left (1 - 2 x\right )^{\frac {5}{2}}}{5000} - \frac {45473 \left (1 - 2 x\right )^{\frac {3}{2}}}{5000} + \frac {2 \sqrt {1 - 2 x}}{3125} + \frac {22 \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 )}{3125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.66, size = 106, normalized size = 1.12 \begin {gather*} \frac {9}{40} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {2889}{1400} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {34371}{5000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {45473}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15625} \, \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 {2}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 66, normalized size = 0.69 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{3125}-\frac {45473\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {34371\,{\left (1-2\,x\right )}^{5/2}}{5000}-\frac {2889\,{\left (1-2\,x\right )}^{7/2}}{1400}+\frac {9\,{\left (1-2\,x\right )}^{9/2}}{40}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{15625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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