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 = 0.0141306, antiderivative size = 68, normalized size of antiderivative = 1., 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} \[ -\frac{67 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{\sqrt{1-2 x}}{42 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
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*}
Mathematica [A] time = 0.0281313, size = 53, normalized size = 0.78 \[ -\frac{\sqrt{1-2 x} (201 x+127)}{294 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 0.7 \begin{align*} -36\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{67\, \left ( 1-2\,x \right ) ^{3/2}}{1764}}+{\frac{65\,\sqrt{1-2\,x}}{756}} \right ) }-{\frac{67\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6504, size = 100, normalized size = 1.47 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39952, size = 196, normalized size = 2.88 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39893, size = 92, normalized size = 1.35 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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