Optimal. Leaf size=108 \[ -\frac {125}{132} (1-2 x)^{11/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {5135}{756} (1-2 x)^{7/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {14}{729} (1-2 x)^{3/2}-\frac {98}{729} \sqrt {1-2 x}+\frac {98}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.04, 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 = {88, 50, 63, 206} \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 50
Rule 63
Rule 88
Rule 206
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} \operatorname {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 = 0.09, size = 68, normalized size = 0.63 \begin {gather*} \frac {\sqrt {1-2 x} \left (8505000 x^5+913500 x^4-7838550 x^3-249219 x^2+3024349 x-830656\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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IntegrateAlgebraic [A] time = 0.08, size = 101, normalized size = 0.94 \begin {gather*} \frac {-1063125 (1-2 x)^{11/2}+5544000 (1-2 x)^{9/2}-7625475 (1-2 x)^{7/2}-5544 (1-2 x)^{5/2}-21560 (1-2 x)^{3/2}-150920 \sqrt {1-2 x}}{1122660}+\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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fricas [A] time = 1.37, 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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giac [A] time = 0.97, 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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maple [A] time = 0.01, size = 74, normalized size = 0.69 \begin {gather*} \frac {98 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2187}-\frac {14 \left (-2 x +1\right )^{\frac {3}{2}}}{729}-\frac {2 \left (-2 x +1\right )^{\frac {5}{2}}}{405}-\frac {5135 \left (-2 x +1\right )^{\frac {7}{2}}}{756}+\frac {400 \left (-2 x +1\right )^{\frac {9}{2}}}{81}-\frac {125 \left (-2 x +1\right )^{\frac {11}{2}}}{132}-\frac {98 \sqrt {-2 x +1}}{729} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, 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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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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sympy [A] time = 76.98, size = 138, normalized size = 1.28 \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}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{729} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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