Optimal. Leaf size=67 \[ -\frac{125}{36} (1-2 x)^{3/2}+\frac{400}{9} \sqrt{1-2 x}+\frac{1331}{28 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]
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Rubi [A] time = 0.0326447, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {87, 43, 63, 206} \[ -\frac{125}{36} (1-2 x)^{3/2}+\frac{400}{9} \sqrt{1-2 x}+\frac{1331}{28 \sqrt{1-2 x}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 87
Rule 43
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\int \left (\frac{1331}{28 (1-2 x)^{3/2}}-\frac{1225}{36 \sqrt{1-2 x}}-\frac{125 x}{6 \sqrt{1-2 x}}-\frac{1}{63 \sqrt{1-2 x} (2+3 x)}\right ) \, dx\\ &=\frac{1331}{28 \sqrt{1-2 x}}+\frac{1225}{36} \sqrt{1-2 x}-\frac{1}{63} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{125}{6} \int \frac{x}{\sqrt{1-2 x}} \, dx\\ &=\frac{1331}{28 \sqrt{1-2 x}}+\frac{1225}{36} \sqrt{1-2 x}+\frac{1}{63} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{125}{6} \int \left (\frac{1}{2 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}\right ) \, dx\\ &=\frac{1331}{28 \sqrt{1-2 x}}+\frac{400}{9} \sqrt{1-2 x}-\frac{125}{36} (1-2 x)^{3/2}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0227722, size = 45, normalized size = 0.67 \[ \frac{-2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-35 \left (75 x^2+405 x-478\right )}{189 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 47, normalized size = 0.7 \begin{align*} -{\frac{125}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{1323}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{1331}{28}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{400}{9}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52665, size = 86, normalized size = 1.28 \begin{align*} -\frac{125}{36} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{1323} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{400}{9} \, \sqrt{-2 \, x + 1} + \frac{1331}{28 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54805, size = 184, normalized size = 2.75 \begin{align*} \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 (875 \, x^{2} + 4725 \, x - 5576\right )} \sqrt{-2 \, x + 1}}{1323 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 38.5807, size = 102, normalized size = 1.52 \begin{align*} - \frac{125 \left (1 - 2 x\right )^{\frac{3}{2}}}{36} + \frac{400 \sqrt{1 - 2 x}}{9} - \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{63} + \frac{1331}{28 \sqrt{1 - 2 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.60467, size = 90, normalized size = 1.34 \begin{align*} -\frac{125}{36} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{1323} \, \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{400}{9} \, \sqrt{-2 \, x + 1} + \frac{1331}{28 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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