Optimal. Leaf size=68 \[ -\frac {67 \sqrt {1-2 x}}{294 (3 x+2)}+\frac {\sqrt {1-2 x}}{42 (3 x+2)^2}-\frac {67 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]
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Rubi [A] time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \begin {gather*} -\frac {67 \sqrt {1-2 x}}{294 (3 x+2)}+\frac {\sqrt {1-2 x}}{42 (3 x+2)^2}-\frac {67 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^3} \, dx &=\frac {\sqrt {1-2 x}}{42 (2+3 x)^2}+\frac {67}{42} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{42 (2+3 x)^2}-\frac {67 \sqrt {1-2 x}}{294 (2+3 x)}+\frac {67}{294} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{42 (2+3 x)^2}-\frac {67 \sqrt {1-2 x}}{294 (2+3 x)}-\frac {67}{294} \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}}{42 (2+3 x)^2}-\frac {67 \sqrt {1-2 x}}{294 (2+3 x)}-\frac {67 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 62, normalized size = 0.91 \begin {gather*} \frac {\sqrt {1-2 x} \left (-\frac {21 (201 x+127)}{(3 x+2)^2}-\frac {134 \sqrt {21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )}{\sqrt {2 x-1}}\right )}{6174} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 61, normalized size = 0.90 \begin {gather*} \frac {(201 (1-2 x)-455) \sqrt {1-2 x}}{147 (3 (1-2 x)-7)^2}-\frac {67 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.53, size = 69, normalized size = 1.01 \begin {gather*} \frac {67 \, \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 (201 \, x + 127\right )} \sqrt {-2 \, x + 1}}{6174 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 68, normalized size = 1.00 \begin {gather*} \frac {67}{6174} \, \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 {201 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 455 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.71 \begin {gather*} -\frac {67 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{3087}-\frac {36 \left (-\frac {67 \left (-2 x +1\right )^{\frac {3}{2}}}{1764}+\frac {65 \sqrt {-2 x +1}}{756}\right )}{\left (-6 x -4\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 74, normalized size = 1.09 \begin {gather*} \frac {67}{6174} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {201 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 455 \, \sqrt {-2 \, x + 1}}{147 \, {\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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mupad [B] time = 0.06, size = 54, normalized size = 0.79 \begin {gather*} -\frac {67\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3087}-\frac {\frac {65\,\sqrt {1-2\,x}}{189}-\frac {67\,{\left (1-2\,x\right )}^{3/2}}{441}}{\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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sympy [F(-1)] 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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