Optimal. Leaf size=61 \[ -\frac{(1-2 x)^{3/2}}{55 (5 x+3)}+\frac{64}{275} \sqrt{1-2 x}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]
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Rubi [A] time = 0.0129792, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 50, 63, 206} \[ -\frac{(1-2 x)^{3/2}}{55 (5 x+3)}+\frac{64}{275} \sqrt{1-2 x}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)}{(3+5 x)^2} \, dx &=-\frac{(1-2 x)^{3/2}}{55 (3+5 x)}+\frac{32}{55} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{64}{275} \sqrt{1-2 x}-\frac{(1-2 x)^{3/2}}{55 (3+5 x)}+\frac{32}{25} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{64}{275} \sqrt{1-2 x}-\frac{(1-2 x)^{3/2}}{55 (3+5 x)}-\frac{32}{25} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{64}{275} \sqrt{1-2 x}-\frac{(1-2 x)^{3/2}}{55 (3+5 x)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0230207, size = 53, normalized size = 0.87 \[ \frac{\sqrt{1-2 x} (30 x+17)}{25 (5 x+3)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 45, normalized size = 0.7 \begin{align*}{\frac{6}{25}\sqrt{1-2\,x}}+{\frac{2}{125}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{64\,\sqrt{55}}{1375}{\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 = 2.01089, size = 84, normalized size = 1.38 \begin{align*} \frac{32}{1375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55594, size = 169, normalized size = 2.77 \begin{align*} \frac{32 \, \sqrt{55}{\left (5 \, x + 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (30 \, x + 17\right )} \sqrt{-2 \, x + 1}}{1375 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 50.9999, size = 178, normalized size = 2.92 \begin{align*} \frac{6 \sqrt{1 - 2 x}}{25} - \frac{44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{25} + \frac{62 \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 )}{25} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32683, size = 88, normalized size = 1.44 \begin{align*} \frac{32}{1375} \, \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{6}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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