Optimal. Leaf size=96 \[ -\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac {365}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 52, 65,
212} \begin {gather*} \frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac {73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac {365}{567} (1-2 x)^{3/2}-\frac {365}{81} \sqrt {1-2 x}+\frac {365}{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 43
Rule 52
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx &=\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}+\frac {73}{42} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac {365}{126} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac {365}{54} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac {2555}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac {2555}{162} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac {365}{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 = 0.15, size = 63, normalized size = 0.66 \begin {gather*} \frac {1}{486} \left (\frac {3 \sqrt {1-2 x} \left (-3521-8731 x-4584 x^2+720 x^3\right )}{(2+3 x)^2}+730 \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 = 0.10, size = 66, normalized size = 0.69
method | result | size |
risch | \(-\frac {1440 x^{4}-9888 x^{3}-12878 x^{2}+1689 x +3521}{162 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {365 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(56\) |
derivativedivides | \(-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {32 \sqrt {1-2 x}}{9}-\frac {28 \left (-\frac {79 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {539 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {365 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(66\) |
default | \(-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {32 \sqrt {1-2 x}}{9}-\frac {28 \left (-\frac {79 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {539 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {365 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(66\) |
trager | \(\frac {\left (720 x^{3}-4584 x^{2}-8731 x -3521\right ) \sqrt {1-2 x}}{162 \left (2+3 x \right )^{2}}+\frac {365 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{486}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 92, normalized size = 0.96 \begin {gather*} -\frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {365}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {32}{9} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (237 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \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 = 1.19, size = 86, normalized size = 0.90 \begin {gather*} \frac {365 \, \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 (720 \, x^{3} - 4584 \, x^{2} - 8731 \, x - 3521\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 = 0.56, size = 86, normalized size = 0.90 \begin {gather*} -\frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {365}{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 {32}{9} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (237 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \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 = 1.21, size = 74, normalized size = 0.77 \begin {gather*} -\frac {32\,\sqrt {1-2\,x}}{9}-\frac {20\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {\frac {3773\,\sqrt {1-2\,x}}{729}-\frac {553\,{\left (1-2\,x\right )}^{3/2}}{243}}{\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 )\,365{}\mathrm {i}}{243} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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