Optimal. Leaf size=109 \[ \frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {1055}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 52, 65,
212} \begin {gather*} \frac {143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac {211}{441} (1-2 x)^{5/2}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {1055}{81} \sqrt {1-2 x}-\frac {1055}{81} \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 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {(1-2 x)^{5/2} (557+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{294} \int \frac {(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{126} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{54} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {7385}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {7385}{162} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {1055}{81} \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 = 0.16, size = 68, normalized size = 0.62 \begin {gather*} \frac {1}{486} \left (\frac {3 \sqrt {1-2 x} \left (10007+25987 x+12828 x^2-3960 x^3+2160 x^4\right )}{(2+3 x)^2}-2110 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 75, normalized size = 0.69
method | result | size |
risch | \(-\frac {4320 x^{5}-10080 x^{4}+29616 x^{3}+39146 x^{2}-5973 x -10007}{162 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {1055 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(61\) |
derivativedivides | \(\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {130 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {1006 \sqrt {1-2 x}}{81}+\frac {-\frac {1043 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2401 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {1055 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(75\) |
default | \(\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {130 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {1006 \sqrt {1-2 x}}{81}+\frac {-\frac {1043 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2401 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {1055 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(75\) |
trager | \(\frac {\left (2160 x^{4}-3960 x^{3}+12828 x^{2}+25987 x +10007\right ) \sqrt {1-2 x}}{162 \left (2+3 x \right )^{2}}-\frac {1055 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{486}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 101, normalized size = 0.93 \begin {gather*} \frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.68, size = 90, normalized size = 0.83 \begin {gather*} \frac {1055 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \, {\left (2160 \, x^{4} - 3960 \, x^{3} + 12828 \, x^{2} + 25987 \, x + 10007\right )} \sqrt {-2 \, x + 1}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 0.64, size = 102, normalized size = 0.94 \begin {gather*} \frac {10}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \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 {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{324 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 82, normalized size = 0.75 \begin {gather*} \frac {1006\,\sqrt {1-2\,x}}{81}+\frac {130\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {10\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {\frac {2401\,\sqrt {1-2\,x}}{729}-\frac {1043\,{\left (1-2\,x\right )}^{3/2}}{729}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,1055{}\mathrm {i}}{243} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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