Optimal. Leaf size=82 \[ \frac {2}{625} \sqrt {1-2 x}-\frac {1299}{500} (1-2 x)^{3/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {27}{140} (1-2 x)^{7/2}-\frac {2}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 82, 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 {27}{140} (1-2 x)^{7/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {1299}{500} (1-2 x)^{3/2}+\frac {2}{625} \sqrt {1-2 x}-\frac {2}{625} \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 90
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac {3897}{500} \sqrt {1-2 x}-\frac {162}{25} (1-2 x)^{3/2}+\frac {27}{20} (1-2 x)^{5/2}+\frac {\sqrt {1-2 x}}{125 (3+5 x)}\right ) \, dx\\ &=-\frac {1299}{500} (1-2 x)^{3/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {27}{140} (1-2 x)^{7/2}+\frac {1}{125} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {2}{625} \sqrt {1-2 x}-\frac {1299}{500} (1-2 x)^{3/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {27}{140} (1-2 x)^{7/2}+\frac {11}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {2}{625} \sqrt {1-2 x}-\frac {1299}{500} (1-2 x)^{3/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {27}{140} (1-2 x)^{7/2}-\frac {11}{625} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2}{625} \sqrt {1-2 x}-\frac {1299}{500} (1-2 x)^{3/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {27}{140} (1-2 x)^{7/2}-\frac {2}{625} \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 = 0.07, size = 56, normalized size = 0.68 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (-6526+5115 x+12555 x^2+6750 x^3\right )-14 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{21875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 56, normalized size = 0.68
method | result | size |
risch | \(-\frac {\left (6750 x^{3}+12555 x^{2}+5115 x -6526\right ) \left (-1+2 x \right )}{4375 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}\) | \(49\) |
derivativedivides | \(-\frac {1299 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {162 \left (1-2 x \right )^{\frac {5}{2}}}{125}-\frac {27 \left (1-2 x \right )^{\frac {7}{2}}}{140}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}+\frac {2 \sqrt {1-2 x}}{625}\) | \(56\) |
default | \(-\frac {1299 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {162 \left (1-2 x \right )^{\frac {5}{2}}}{125}-\frac {27 \left (1-2 x \right )^{\frac {7}{2}}}{140}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3125}+\frac {2 \sqrt {1-2 x}}{625}\) | \(56\) |
trager | \(\left (\frac {54}{35} x^{3}+\frac {2511}{875} x^{2}+\frac {1023}{875} x -\frac {6526}{4375}\right ) \sqrt {1-2 x}-\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{3125}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 73, normalized size = 0.89 \begin {gather*} -\frac {27}{140} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {162}{125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1299}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.73, size = 61, normalized size = 0.74 \begin {gather*} \frac {1}{3125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{4375} \, {\left (6750 \, x^{3} + 12555 \, x^{2} + 5115 \, x - 6526\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.98, size = 107, normalized size = 1.30 \begin {gather*} - \frac {27 \left (1 - 2 x\right )^{\frac {7}{2}}}{140} + \frac {162 \left (1 - 2 x\right )^{\frac {5}{2}}}{125} - \frac {1299 \left (1 - 2 x\right )^{\frac {3}{2}}}{500} + \frac {2 \sqrt {1 - 2 x}}{625} + \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 )}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 90, normalized size = 1.10 \begin {gather*} \frac {27}{140} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {162}{125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1299}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{3125} \, \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}{625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 57, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{625}-\frac {1299\,{\left (1-2\,x\right )}^{3/2}}{500}+\frac {162\,{\left (1-2\,x\right )}^{5/2}}{125}-\frac {27\,{\left (1-2\,x\right )}^{7/2}}{140}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{3125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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