Optimal. Leaf size=68 \[ -\frac{(1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{50 (5 x+3)}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]
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Rubi [A] time = 0.0140419, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {47, 63, 206} \[ -\frac{(1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{50 (5 x+3)}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(3+5 x)^3} \, dx &=-\frac{(1-2 x)^{3/2}}{10 (3+5 x)^2}-\frac{3}{10} \int \frac{\sqrt{1-2 x}}{(3+5 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac{3 \sqrt{1-2 x}}{50 (3+5 x)}+\frac{3}{50} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac{3 \sqrt{1-2 x}}{50 (3+5 x)}-\frac{3}{50} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{(1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac{3 \sqrt{1-2 x}}{50 (3+5 x)}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0406294, size = 58, normalized size = 0.85 \[ \frac{-50 x^2+17 x+4}{50 \sqrt{1-2 x} (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 48, normalized size = 0.7 \begin{align*} 200\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{3/2}}{200}}+{\frac{33\,\sqrt{1-2\,x}}{5000}} \right ) }-{\frac{3\,\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 = 1.82484, size = 100, normalized size = 1.47 \begin{align*} \frac{3}{2750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{25 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 33 \, \sqrt{-2 \, x + 1}}{25 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53438, size = 193, normalized size = 2.84 \begin{align*} \frac{3 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (25 \, x + 4\right )} \sqrt{-2 \, x + 1}}{2750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.48886, size = 235, normalized size = 3.46 \begin{align*} \begin{cases} - \frac{3 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1375} - \frac{\sqrt{2}}{50 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{77 \sqrt{2}}{2500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{121 \sqrt{2}}{12500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\\frac{3 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1375} + \frac{\sqrt{2} i}{50 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{77 \sqrt{2} i}{2500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{121 \sqrt{2} i}{12500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.99327, size = 92, normalized size = 1.35 \begin{align*} \frac{3}{2750} \, \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{25 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 33 \, \sqrt{-2 \, x + 1}}{100 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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