Optimal. Leaf size=82 \[ \frac{9}{70} (1-2 x)^{7/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{2}{375} (1-2 x)^{3/2}+\frac{22}{625} \sqrt{1-2 x}-\frac{22}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.029403, antiderivative size = 82, 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 = {88, 50, 63, 206} \[ \frac{9}{70} (1-2 x)^{7/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{2}{375} (1-2 x)^{3/2}+\frac{22}{625} \sqrt{1-2 x}-\frac{22}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 88
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx &=\int \left (\frac{111}{50} (1-2 x)^{3/2}-\frac{9}{10} (1-2 x)^{5/2}+\frac{(1-2 x)^{3/2}}{25 (3+5 x)}\right ) \, dx\\ &=-\frac{111}{250} (1-2 x)^{5/2}+\frac{9}{70} (1-2 x)^{7/2}+\frac{1}{25} \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac{2}{375} (1-2 x)^{3/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{9}{70} (1-2 x)^{7/2}+\frac{11}{125} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{22}{625} \sqrt{1-2 x}+\frac{2}{375} (1-2 x)^{3/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{9}{70} (1-2 x)^{7/2}+\frac{121}{625} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{22}{625} \sqrt{1-2 x}+\frac{2}{375} (1-2 x)^{3/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{9}{70} (1-2 x)^{7/2}-\frac{121}{625} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{22}{625} \sqrt{1-2 x}+\frac{2}{375} (1-2 x)^{3/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{9}{70} (1-2 x)^{7/2}-\frac{22}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0415151, size = 56, normalized size = 0.68 \[ \frac{-5 \sqrt{1-2 x} \left (13500 x^3+3060 x^2-13045 x+3608\right )-462 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{65625} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 56, normalized size = 0.7 \begin{align*}{\frac{2}{375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{111}{250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{9}{70} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{22\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{625}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74799, size = 99, normalized size = 1.21 \begin{align*} \frac{9}{70} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{111}{250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{625} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3829, size = 203, normalized size = 2.48 \begin{align*} \frac{11}{3125} \, \sqrt{11} \sqrt{5} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac{1}{13125} \,{\left (13500 \, x^{3} + 3060 \, x^{2} - 13045 \, x + 3608\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.7788, size = 114, normalized size = 1.39 \begin{align*} \frac{9 \left (1 - 2 x\right )^{\frac{7}{2}}}{70} - \frac{111 \left (1 - 2 x\right )^{\frac{5}{2}}}{250} + \frac{2 \left (1 - 2 x\right )^{\frac{3}{2}}}{375} + \frac{22 \sqrt{1 - 2 x}}{625} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{625} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.58731, size = 122, normalized size = 1.49 \begin{align*} -\frac{9}{70} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{111}{250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{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{22}{625} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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