Optimal. Leaf size=81 \[ \frac{(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac{71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac{71}{63} \sqrt{1-2 x}+\frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]
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Rubi [A] time = 0.0173287, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 50, 63, 206} \[ \frac{(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac{71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac{71}{63} \sqrt{1-2 x}+\frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx &=\frac{(1-2 x)^{5/2}}{42 (2+3 x)^2}+\frac{71}{42} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac{71 (1-2 x)^{3/2}}{126 (2+3 x)}-\frac{71}{42} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=-\frac{71}{63} \sqrt{1-2 x}+\frac{(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac{71 (1-2 x)^{3/2}}{126 (2+3 x)}-\frac{71}{18} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{71}{63} \sqrt{1-2 x}+\frac{(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac{71 (1-2 x)^{3/2}}{126 (2+3 x)}+\frac{71}{18} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{71}{63} \sqrt{1-2 x}+\frac{(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac{71 (1-2 x)^{3/2}}{126 (2+3 x)}+\frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0175819, size = 48, normalized size = 0.59 \[ \frac{(1-2 x)^{5/2} \left (245-284 (3 x+2)^2 \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{10290 (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 57, normalized size = 0.7 \begin{align*} -{\frac{20}{27}\sqrt{1-2\,x}}-{\frac{4}{3\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{25}{4} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{511}{36}\sqrt{1-2\,x}} \right ) }+{\frac{71\,\sqrt{21}}{189}{\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 = 1.72785, size = 112, normalized size = 1.38 \begin{align*} -\frac{71}{378} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} + \frac{225 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 511 \, \sqrt{-2 \, x + 1}}{27 \,{\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 = 1.32964, size = 208, normalized size = 2.57 \begin{align*} \frac{71 \, \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 (120 \, x^{2} + 235 \, x + 101\right )} \sqrt{-2 \, x + 1}}{378 \,{\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.85221, size = 104, normalized size = 1.28 \begin{align*} -\frac{71}{378} \, \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{20}{27} \, \sqrt{-2 \, x + 1} + \frac{225 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 511 \, \sqrt{-2 \, x + 1}}{108 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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