Optimal. Leaf size=67 \[ -\frac{27}{100} (1-2 x)^{5/2}+\frac{54}{25} (1-2 x)^{3/2}-\frac{3897}{500} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]
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Rubi [A] time = 0.0225344, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {88, 63, 206} \[ -\frac{27}{100} (1-2 x)^{5/2}+\frac{54}{25} (1-2 x)^{3/2}-\frac{3897}{500} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3}{\sqrt{1-2 x} (3+5 x)} \, dx &=\int \left (\frac{3897}{500 \sqrt{1-2 x}}-\frac{162}{25} \sqrt{1-2 x}+\frac{27}{20} (1-2 x)^{3/2}+\frac{1}{125 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=-\frac{3897}{500} \sqrt{1-2 x}+\frac{54}{25} (1-2 x)^{3/2}-\frac{27}{100} (1-2 x)^{5/2}+\frac{1}{125} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{3897}{500} \sqrt{1-2 x}+\frac{54}{25} (1-2 x)^{3/2}-\frac{27}{100} (1-2 x)^{5/2}-\frac{1}{125} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{3897}{500} \sqrt{1-2 x}+\frac{54}{25} (1-2 x)^{3/2}-\frac{27}{100} (1-2 x)^{5/2}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0284061, size = 51, normalized size = 0.76 \[ \frac{-495 \sqrt{1-2 x} \left (15 x^2+45 x+82\right )-2 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6875} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 47, normalized size = 0.7 \begin{align*}{\frac{54}{25} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{27}{100} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{55}}{6875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{3897}{500}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78801, size = 86, normalized size = 1.28 \begin{align*} -\frac{27}{100} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{54}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{6875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3897}{500} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66623, size = 153, normalized size = 2.28 \begin{align*} -\frac{9}{125} \,{\left (15 \, x^{2} + 45 \, x + 82\right )} \sqrt{-2 \, x + 1} + \frac{1}{6875} \, \sqrt{55} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.9137, size = 102, normalized size = 1.52 \begin{align*} - \frac{27 \left (1 - 2 x\right )^{\frac{5}{2}}}{100} + \frac{54 \left (1 - 2 x\right )^{\frac{3}{2}}}{25} - \frac{3897 \sqrt{1 - 2 x}}{500} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{1 - 2 x}} \right )}}{55} & \text{for}\: \frac{1}{1 - 2 x} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{1 - 2 x}} \right )}}{55} & \text{for}\: \frac{1}{1 - 2 x} < \frac{5}{11} \end{cases}\right )}{125} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.92683, size = 100, normalized size = 1.49 \begin{align*} -\frac{27}{100} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{54}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{6875} \, \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{3897}{500} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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