Optimal. Leaf size=121 \[ \frac {31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac {(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac {4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac {24965 \sqrt {1-2 x}}{15876}-\frac {24965 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {89, 78, 47, 50, 63, 206} \begin {gather*} \frac {31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac {(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac {4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac {24965 \sqrt {1-2 x}}{15876}-\frac {24965 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {1}{252} \int \frac {(1-2 x)^{5/2} (1121+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}+\frac {4993 \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3} \, dx}{1764}\\ &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}-\frac {24965 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}+\frac {24965 \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx}{10584}\\ &=\frac {24965 \sqrt {1-2 x}}{15876}-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}+\frac {24965 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{4536}\\ &=\frac {24965 \sqrt {1-2 x}}{15876}-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}-\frac {24965 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{4536}\\ &=\frac {24965 \sqrt {1-2 x}}{15876}-\frac {(1-2 x)^{7/2}}{252 (2+3 x)^4}+\frac {31 (1-2 x)^{7/2}}{588 (2+3 x)^3}-\frac {4993 (1-2 x)^{5/2}}{10584 (2+3 x)^2}+\frac {24965 (1-2 x)^{3/2}}{31752 (2+3 x)}-\frac {24965 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 47, normalized size = 0.39 \begin {gather*} \frac {(1-2 x)^{7/2} \left (\frac {2401 (279 x+179)}{(3 x+2)^4}-39944 \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{4235364} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 88, normalized size = 0.73 \begin {gather*} \frac {\left (151200 (1-2 x)^4-1835865 (1-2 x)^3+7654269 (1-2 x)^2-13456135 (1-2 x)+8562995\right ) \sqrt {1-2 x}}{2268 (3 (1-2 x)-7)^4}-\frac {24965 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.58, size = 104, normalized size = 0.86 \begin {gather*} \frac {24965 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (302400 \, x^{4} + 1231065 \, x^{3} + 1526937 \, x^{2} + 762598 \, x + 134558\right )} \sqrt {-2 \, x + 1}}{95256 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 109, normalized size = 0.90 \begin {gather*} \frac {24965}{95256} \, \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 {200}{243} \, \sqrt {-2 \, x + 1} + \frac {1273995 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 8145207 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 17318805 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 12243385 \, \sqrt {-2 \, x + 1}}{108864 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 75, normalized size = 0.62 \begin {gather*} -\frac {24965 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{47628}+\frac {200 \sqrt {-2 x +1}}{243}+\frac {-\frac {47185 \left (-2 x +1\right )^{\frac {7}{2}}}{252}+\frac {129289 \left (-2 x +1\right )^{\frac {5}{2}}}{108}-\frac {824705 \left (-2 x +1\right )^{\frac {3}{2}}}{324}+\frac {1749055 \sqrt {-2 x +1}}{972}}{\left (-6 x -4\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 119, normalized size = 0.98 \begin {gather*} \frac {24965}{95256} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {200}{243} \, \sqrt {-2 \, x + 1} - \frac {1273995 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 8145207 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 17318805 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12243385 \, \sqrt {-2 \, x + 1}}{6804 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 98, normalized size = 0.81 \begin {gather*} \frac {200\,\sqrt {1-2\,x}}{243}-\frac {24965\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{47628}+\frac {\frac {1749055\,\sqrt {1-2\,x}}{78732}-\frac {824705\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {129289\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {47185\,{\left (1-2\,x\right )}^{7/2}}{20412}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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