Optimal. Leaf size=93 \[ \frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2}+\frac {65 \sqrt {1-2 x}}{6 (2+3 x)}+\frac {2243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}-22 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 93, 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 = {100, 156, 162,
65, 212} \begin {gather*} \frac {65 \sqrt {1-2 x}}{6 (3 x+2)}+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2}+\frac {2243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}-22 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)} \, dx &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {87-97 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2}+\frac {65 \sqrt {1-2 x}}{6 (2+3 x)}+\frac {1}{42} \int \frac {3717-2275 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2}+\frac {65 \sqrt {1-2 x}}{6 (2+3 x)}-\frac {2243}{6} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+605 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2}+\frac {65 \sqrt {1-2 x}}{6 (2+3 x)}+\frac {2243}{6} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-605 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2}+\frac {65 \sqrt {1-2 x}}{6 (2+3 x)}+\frac {2243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}-22 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 78, normalized size = 0.84 \begin {gather*} \frac {\sqrt {1-2 x} (137+195 x)}{6 (2+3 x)^2}+\frac {2243 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}-22 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 66, normalized size = 0.71
method | result | size |
risch | \(-\frac {390 x^{2}+79 x -137}{6 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {2243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{63}\) | \(64\) |
derivativedivides | \(-22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {18 \left (\frac {65 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {469 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{2}}+\frac {2243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{63}\) | \(66\) |
default | \(-22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {18 \left (\frac {65 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {469 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{2}}+\frac {2243 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{63}\) | \(66\) |
trager | \(\frac {\left (195 x +137\right ) \sqrt {1-2 x}}{6 \left (2+3 x \right )^{2}}-\frac {2243 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{126}+11 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 110, normalized size = 1.18 \begin {gather*} 11 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2243}{126} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {195 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 469 \, \sqrt {-2 \, x + 1}}{3 \, {\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 = 0.66, size = 110, normalized size = 1.18 \begin {gather*} \frac {1386 \, \sqrt {55} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 2243 \, \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 (195 \, x + 137\right )} \sqrt {-2 \, x + 1}}{126 \, {\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 = 0.67, size = 107, normalized size = 1.15 \begin {gather*} 11 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2243}{126} \, \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 {195 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 469 \, \sqrt {-2 \, x + 1}}{12 \, {\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 = 0.09, size = 71, normalized size = 0.76 \begin {gather*} \frac {2243\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{63}-22\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {469\,\sqrt {1-2\,x}}{27}-\frac {65\,{\left (1-2\,x\right )}^{3/2}}{9}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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