Optimal. Leaf size=89 \[ -\frac{(1-2 x)^{5/2}}{275 (5 x+3)}-\frac{9}{125} (1-2 x)^{5/2}+\frac{42 (1-2 x)^{3/2}}{1375}+\frac{126}{625} \sqrt{1-2 x}-\frac{126}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0248406, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 80, 50, 63, 206} \[ -\frac{(1-2 x)^{5/2}}{275 (5 x+3)}-\frac{9}{125} (1-2 x)^{5/2}+\frac{42 (1-2 x)^{3/2}}{1375}+\frac{126}{625} \sqrt{1-2 x}-\frac{126}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 89
Rule 80
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^2} \, dx &=-\frac{(1-2 x)^{5/2}}{275 (3+5 x)}+\frac{1}{275} \int \frac{(1-2 x)^{3/2} (360+495 x)}{3+5 x} \, dx\\ &=-\frac{9}{125} (1-2 x)^{5/2}-\frac{(1-2 x)^{5/2}}{275 (3+5 x)}+\frac{63}{275} \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac{42 (1-2 x)^{3/2}}{1375}-\frac{9}{125} (1-2 x)^{5/2}-\frac{(1-2 x)^{5/2}}{275 (3+5 x)}+\frac{63}{125} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{126}{625} \sqrt{1-2 x}+\frac{42 (1-2 x)^{3/2}}{1375}-\frac{9}{125} (1-2 x)^{5/2}-\frac{(1-2 x)^{5/2}}{275 (3+5 x)}+\frac{693}{625} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{126}{625} \sqrt{1-2 x}+\frac{42 (1-2 x)^{3/2}}{1375}-\frac{9}{125} (1-2 x)^{5/2}-\frac{(1-2 x)^{5/2}}{275 (3+5 x)}-\frac{693}{625} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{126}{625} \sqrt{1-2 x}+\frac{42 (1-2 x)^{3/2}}{1375}-\frac{9}{125} (1-2 x)^{5/2}-\frac{(1-2 x)^{5/2}}{275 (3+5 x)}-\frac{126}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0356058, size = 63, normalized size = 0.71 \[ \frac{\frac{5 \sqrt{1-2 x} \left (-900 x^3+160 x^2+935 x+298\right )}{5 x+3}-126 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 63, normalized size = 0.7 \begin{align*} -{\frac{9}{125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{4}{125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{128}{625}\sqrt{1-2\,x}}+{\frac{22}{3125}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{126\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.89637, size = 108, normalized size = 1.21 \begin{align*} -\frac{9}{125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{4}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{63}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{128}{625} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5757, size = 219, normalized size = 2.46 \begin{align*} \frac{63 \, \sqrt{11} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \,{\left (900 \, x^{3} - 160 \, x^{2} - 935 \, x - 298\right )} \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\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 = 1.96852, size = 122, normalized size = 1.37 \begin{align*} -\frac{9}{125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{4}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{63}{3125} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{128}{625} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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