Optimal. Leaf size=48 \[ \frac {\sqrt {1-2 x}}{21 (3 x+2)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {78, 63, 206} \[ \frac {\sqrt {1-2 x}}{21 (3 x+2)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^2} \, dx &=\frac {\sqrt {1-2 x}}{21 (2+3 x)}+\frac {34}{21} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{21 (2+3 x)}-\frac {34}{21} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x}}{21 (2+3 x)}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.94 \[ \frac {\sqrt {1-2 x}}{63 x+42}-\frac {68 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 54, normalized size = 1.12 \[ \frac {34 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, \sqrt {-2 \, x + 1}}{441 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 56, normalized size = 1.17 \[ \frac {34}{441} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {\sqrt {-2 \, x + 1}}{21 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.75 \[ -\frac {68 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{441}-\frac {2 \sqrt {-2 x +1}}{63 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 53, normalized size = 1.10 \[ \frac {34}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {\sqrt {-2 \, x + 1}}{21 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 35, normalized size = 0.73 \[ \frac {2\,\sqrt {1-2\,x}}{63\,\left (2\,x+\frac {4}{3}\right )}-\frac {68\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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