Optimal. Leaf size=54 \[ \frac {121}{14 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {89, 65, 212}
\begin {gather*} \frac {25}{6} \sqrt {1-2 x}+\frac {121}{14 \sqrt {1-2 x}}-\frac {2 \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 89
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\int \left (\frac {121}{14 (1-2 x)^{3/2}}-\frac {25}{6 \sqrt {1-2 x}}+\frac {1}{21 \sqrt {1-2 x} (2+3 x)}\right ) \, dx\\ &=\frac {121}{14 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {1-2 x}+\frac {1}{21} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {1-2 x}-\frac {1}{21} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {121}{14 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 46, normalized size = 0.85 \begin {gather*} \frac {269-175 x}{21 \sqrt {1-2 x}}-\frac {2 \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 = 0.13, size = 38, normalized size = 0.70
method | result | size |
risch | \(-\frac {175 x -269}{21 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(34\) |
derivativedivides | \(-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}+\frac {121}{14 \sqrt {1-2 x}}+\frac {25 \sqrt {1-2 x}}{6}\) | \(38\) |
default | \(-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}+\frac {121}{14 \sqrt {1-2 x}}+\frac {25 \sqrt {1-2 x}}{6}\) | \(38\) |
trager | \(\frac {\left (175 x -269\right ) \sqrt {1-2 x}}{-21+42 x}-\frac {\RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{441}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 55, normalized size = 1.02 \begin {gather*} \frac {1}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {25}{6} \, \sqrt {-2 \, x + 1} + \frac {121}{14 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.94, size = 58, normalized size = 1.07 \begin {gather*} \frac {\sqrt {21} {\left (2 \, x - 1\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (175 \, x - 269\right )} \sqrt {-2 \, x + 1}}{441 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 19.02, size = 83, normalized size = 1.54 \begin {gather*} \frac {25 \sqrt {1 - 2 x}}{6} + \frac {2 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{21} + \frac {121}{14 \sqrt {1 - 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 58, normalized size = 1.07 \begin {gather*} \frac {1}{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}{6} \, \sqrt {-2 \, x + 1} + \frac {121}{14 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 37, normalized size = 0.69 \begin {gather*} \frac {121}{14\,\sqrt {1-2\,x}}-\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}+\frac {25\,\sqrt {1-2\,x}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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