Optimal. Leaf size=108 \[ -\frac {98}{729} \sqrt {1-2 x}-\frac {14}{729} (1-2 x)^{3/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}+\frac {98}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {125}{132} (1-2 x)^{11/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {5135}{756} (1-2 x)^{7/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {14}{729} (1-2 x)^{3/2}-\frac {98}{729} \sqrt {1-2 x}+\frac {98}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \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 {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx &=\int \left (\frac {5135}{108} (1-2 x)^{5/2}-\frac {400}{9} (1-2 x)^{7/2}+\frac {125}{12} (1-2 x)^{9/2}-\frac {(1-2 x)^{5/2}}{27 (2+3 x)}\right ) \, dx\\ &=-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}-\frac {1}{27} \int \frac {(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=-\frac {2}{405} (1-2 x)^{5/2}-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}-\frac {7}{81} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {14}{729} (1-2 x)^{3/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}-\frac {49}{243} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {98}{729} \sqrt {1-2 x}-\frac {14}{729} (1-2 x)^{3/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}-\frac {343}{729} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {98}{729} \sqrt {1-2 x}-\frac {14}{729} (1-2 x)^{3/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}+\frac {343}{729} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {98}{729} \sqrt {1-2 x}-\frac {14}{729} (1-2 x)^{3/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}+\frac {98}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 68, normalized size = 0.63 \begin {gather*} \frac {\sqrt {1-2 x} \left (-830656+3024349 x-249219 x^2-7838550 x^3+913500 x^4+8505000 x^5\right )}{280665}+\frac {98}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 74, normalized size = 0.69
method | result | size |
risch | \(-\frac {\left (8505000 x^{5}+913500 x^{4}-7838550 x^{3}-249219 x^{2}+3024349 x -830656\right ) \left (-1+2 x \right )}{280665 \sqrt {1-2 x}}+\frac {98 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}\) | \(59\) |
derivativedivides | \(-\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {5135 \left (1-2 x \right )^{\frac {7}{2}}}{756}+\frac {400 \left (1-2 x \right )^{\frac {9}{2}}}{81}-\frac {125 \left (1-2 x \right )^{\frac {11}{2}}}{132}+\frac {98 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}-\frac {98 \sqrt {1-2 x}}{729}\) | \(74\) |
default | \(-\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {5135 \left (1-2 x \right )^{\frac {7}{2}}}{756}+\frac {400 \left (1-2 x \right )^{\frac {9}{2}}}{81}-\frac {125 \left (1-2 x \right )^{\frac {11}{2}}}{132}+\frac {98 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}-\frac {98 \sqrt {1-2 x}}{729}\) | \(74\) |
trager | \(\left (\frac {1000}{33} x^{5}+\frac {2900}{891} x^{4}-\frac {174190}{6237} x^{3}-\frac {27691}{31185} x^{2}+\frac {3024349}{280665} x -\frac {830656}{280665}\right ) \sqrt {1-2 x}-\frac {49 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2187}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 91, normalized size = 0.84 \begin {gather*} -\frac {125}{132} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {400}{81} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {5135}{756} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {2}{405} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {14}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{2187} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {98}{729} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 72, normalized size = 0.67 \begin {gather*} \frac {49}{2187} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac {1}{280665} \, {\left (8505000 \, x^{5} + 913500 \, x^{4} - 7838550 \, x^{3} - 249219 \, x^{2} + 3024349 \, x - 830656\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 45.04, size = 131, normalized size = 1.21 \begin {gather*} - \frac {125 \left (1 - 2 x\right )^{\frac {11}{2}}}{132} + \frac {400 \left (1 - 2 x\right )^{\frac {9}{2}}}{81} - \frac {5135 \left (1 - 2 x\right )^{\frac {7}{2}}}{756} - \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{405} - \frac {14 \left (1 - 2 x\right )^{\frac {3}{2}}}{729} - \frac {98 \sqrt {1 - 2 x}}{729} - \frac {686 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{729} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 122, normalized size = 1.13 \begin {gather*} \frac {125}{132} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {400}{81} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {5135}{756} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {2}{405} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {14}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{2187} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {98}{729} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 75, normalized size = 0.69 \begin {gather*} \frac {400\,{\left (1-2\,x\right )}^{9/2}}{81}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{729}-\frac {2\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {5135\,{\left (1-2\,x\right )}^{7/2}}{756}-\frac {98\,\sqrt {1-2\,x}}{729}-\frac {125\,{\left (1-2\,x\right )}^{11/2}}{132}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{2187} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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