Optimal. Leaf size=61 \[ -\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 81, 65, 212}
\begin {gather*} -\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (3 x+2)}+\frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx &=-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {1}{63} \int \frac {281+525 x}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}-\frac {23}{21} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {23}{21} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 51, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {1-2 x} (117+175 x)}{42+63 x}+\frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 45, normalized size = 0.74
method | result | size |
derivativedivides | \(-\frac {25 \sqrt {1-2 x}}{9}+\frac {2 \sqrt {1-2 x}}{189 \left (-\frac {4}{3}-2 x \right )}+\frac {46 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(45\) |
default | \(-\frac {25 \sqrt {1-2 x}}{9}+\frac {2 \sqrt {1-2 x}}{189 \left (-\frac {4}{3}-2 x \right )}+\frac {46 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(45\) |
risch | \(\frac {350 x^{2}+59 x -117}{21 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {46 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(46\) |
trager | \(-\frac {\left (175 x +117\right ) \sqrt {1-2 x}}{21 \left (2+3 x \right )}+\frac {23 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{441}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 62, normalized size = 1.02 \begin {gather*} -\frac {23}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {25}{9} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{63 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.12, size = 60, normalized size = 0.98 \begin {gather*} \frac {23 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (175 \, x + 117\right )} \sqrt {-2 \, x + 1}}{441 \, {\left (3 \, x + 2\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 = 65, normalized size = 1.07 \begin {gather*} -\frac {23}{441} \, \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 {25}{9} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{63 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 44, normalized size = 0.72 \begin {gather*} \frac {46\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}-\frac {2\,\sqrt {1-2\,x}}{189\,\left (2\,x+\frac {4}{3}\right )}-\frac {25\,\sqrt {1-2\,x}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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