Optimal. Leaf size=67 \[ -\frac{9}{20} (1-2 x)^{3/2}+\frac{162}{25} \sqrt{1-2 x}+\frac{343}{44 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]
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Rubi [A] time = 0.0341693, 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} \[ -\frac{9}{20} (1-2 x)^{3/2}+\frac{162}{25} \sqrt{1-2 x}+\frac{343}{44 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 87
Rule 43
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac{343}{44 (1-2 x)^{3/2}}-\frac{513}{100 \sqrt{1-2 x}}-\frac{27 x}{10 \sqrt{1-2 x}}+\frac{1}{275 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac{343}{44 \sqrt{1-2 x}}+\frac{513}{100} \sqrt{1-2 x}+\frac{1}{275} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx-\frac{27}{10} \int \frac{x}{\sqrt{1-2 x}} \, dx\\ &=\frac{343}{44 \sqrt{1-2 x}}+\frac{513}{100} \sqrt{1-2 x}-\frac{1}{275} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{27}{10} \int \left (\frac{1}{2 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}\right ) \, dx\\ &=\frac{343}{44 \sqrt{1-2 x}}+\frac{162}{25} \sqrt{1-2 x}-\frac{9}{20} (1-2 x)^{3/2}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0193607, size = 45, normalized size = 0.67 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-99 \left (25 x^2+155 x-192\right )}{1375 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 47, normalized size = 0.7 \begin{align*} -{\frac{9}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{15125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{343}{44}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{162}{25}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.791, size = 86, normalized size = 1.28 \begin{align*} -\frac{9}{20} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{15125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{162}{25} \, \sqrt{-2 \, x + 1} + \frac{343}{44 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70726, size = 185, normalized size = 2.76 \begin{align*} \frac{\sqrt{55}{\left (2 \, x - 1\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (495 \, x^{2} + 3069 \, x - 3802\right )} \sqrt{-2 \, x + 1}}{15125 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.9039, size = 102, normalized size = 1.52 \begin{align*} - \frac{9 \left (1 - 2 x\right )^{\frac{3}{2}}}{20} + \frac{162 \sqrt{1 - 2 x}}{25} + \frac{2 \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 )}{275} + \frac{343}{44 \sqrt{1 - 2 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.03867, size = 90, normalized size = 1.34 \begin{align*} -\frac{9}{20} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{15125} \, \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{162}{25} \, \sqrt{-2 \, x + 1} + \frac{343}{44 \, \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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