Optimal. Leaf size=118 \[ -\frac {31}{3} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {6 \sqrt {1-2 x} (3+5 x)^3}{2+3 x}+\frac {1}{54} \sqrt {1-2 x} (367+1715 x)+\frac {887 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
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Rubi [A]
time = 0.03, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 154, 158,
152, 65, 212} \begin {gather*} \frac {6 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac {31}{3} \sqrt {1-2 x} (5 x+3)^2+\frac {1}{54} \sqrt {1-2 x} (1715 x+367)+\frac {887 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 99
Rule 152
Rule 154
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {6 \sqrt {1-2 x} (3+5 x)^3}{2+3 x}-\frac {1}{18} \int \frac {(801-2790 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {31}{3} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {6 \sqrt {1-2 x} (3+5 x)^3}{2+3 x}+\frac {1}{270} \int \frac {(3015-15435 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {31}{3} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {6 \sqrt {1-2 x} (3+5 x)^3}{2+3 x}+\frac {1}{54} \sqrt {1-2 x} (367+1715 x)-\frac {887}{54} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {31}{3} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {6 \sqrt {1-2 x} (3+5 x)^3}{2+3 x}+\frac {1}{54} \sqrt {1-2 x} (367+1715 x)+\frac {887}{54} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {31}{3} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {6 \sqrt {1-2 x} (3+5 x)^3}{2+3 x}+\frac {1}{54} \sqrt {1-2 x} (367+1715 x)+\frac {887 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 63, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {1-2 x} \left (1367+2965 x+570 x^2+1800 x^4\right )}{54 (2+3 x)^2}+\frac {887 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 75, normalized size = 0.64
method | result | size |
risch | \(\frac {3600 x^{5}-1800 x^{4}+1140 x^{3}+5360 x^{2}-231 x -1367}{54 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {887 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(61\) |
derivativedivides | \(-\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {50 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {370 \sqrt {1-2 x}}{81}-\frac {2 \left (-\frac {215 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {497 \sqrt {1-2 x}}{18}\right )}{9 \left (-4-6 x \right )^{2}}+\frac {887 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(75\) |
default | \(-\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {50 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {370 \sqrt {1-2 x}}{81}-\frac {2 \left (-\frac {215 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {497 \sqrt {1-2 x}}{18}\right )}{9 \left (-4-6 x \right )^{2}}+\frac {887 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(75\) |
trager | \(-\frac {\left (1800 x^{4}+570 x^{2}+2965 x +1367\right ) \sqrt {1-2 x}}{54 \left (2+3 x \right )^{2}}+\frac {887 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1134}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 101, normalized size = 0.86 \begin {gather*} -\frac {25}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {887}{1134} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {370}{81} \, \sqrt {-2 \, x + 1} + \frac {215 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{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 = 1.01, size = 80, normalized size = 0.68 \begin {gather*} \frac {887 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1800 \, x^{4} + 570 \, x^{2} + 2965 \, x + 1367\right )} \sqrt {-2 \, x + 1}}{1134 \, {\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 = 0.95, size = 102, normalized size = 0.86 \begin {gather*} -\frac {25}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {887}{1134} \, \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 {370}{81} \, \sqrt {-2 \, x + 1} + \frac {215 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{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 = 0.05, size = 83, normalized size = 0.70 \begin {gather*} -\frac {370\,\sqrt {1-2\,x}}{81}-\frac {50\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {25\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {\frac {497\,\sqrt {1-2\,x}}{729}-\frac {215\,{\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 )\,887{}\mathrm {i}}{567} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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