Optimal. Leaf size=116 \[ -\frac{3}{50} \sqrt{5 x+3} (1-2 x)^{5/2}-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{5 x+3}}+\frac{119 \sqrt{5 x+3} (1-2 x)^{3/2}}{2200}+\frac{357 \sqrt{5 x+3} \sqrt{1-2 x}}{2000}+\frac{3927 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]
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Rubi [A] time = 0.0313429, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \[ -\frac{3}{50} \sqrt{5 x+3} (1-2 x)^{5/2}-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{5 x+3}}+\frac{119 \sqrt{5 x+3} (1-2 x)^{3/2}}{2200}+\frac{357 \sqrt{5 x+3} \sqrt{1-2 x}}{2000}+\frac{3927 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{3+5 x}}+\frac{2}{275} \int \frac{(1-2 x)^{3/2} \left (\frac{355}{2}+\frac{495 x}{2}\right )}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{3+5 x}}-\frac{3}{50} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{119}{220} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{3+5 x}}+\frac{119 (1-2 x)^{3/2} \sqrt{3+5 x}}{2200}-\frac{3}{50} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{357}{400} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{3+5 x}}+\frac{357 \sqrt{1-2 x} \sqrt{3+5 x}}{2000}+\frac{119 (1-2 x)^{3/2} \sqrt{3+5 x}}{2200}-\frac{3}{50} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{3927 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{4000}\\ &=-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{3+5 x}}+\frac{357 \sqrt{1-2 x} \sqrt{3+5 x}}{2000}+\frac{119 (1-2 x)^{3/2} \sqrt{3+5 x}}{2200}-\frac{3}{50} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{3927 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{3+5 x}}+\frac{357 \sqrt{1-2 x} \sqrt{3+5 x}}{2000}+\frac{119 (1-2 x)^{3/2} \sqrt{3+5 x}}{2200}-\frac{3}{50} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{3927 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.033615, size = 83, normalized size = 0.72 \[ \frac{10 \left (4800 x^4-2040 x^3-5330 x^2+533 x+1021\right )-3927 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{20000 \sqrt{1-2 x} \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 116, normalized size = 1. \begin{align*}{\frac{1}{40000} \left ( -48000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+19635\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+11781\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +51500\,x\sqrt{-10\,{x}^{2}-x+3}+20420\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 3.75786, size = 208, normalized size = 1.79 \begin{align*} -\frac{11979}{200000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{957}{25000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{125} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{99}{500} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{2277}{10000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{99}{1250} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{125 \,{\left (5 \, x + 3\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50304, size = 265, normalized size = 2.28 \begin{align*} -\frac{3927 \, \sqrt{10}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (2400 \, x^{3} + 180 \, x^{2} - 2575 \, x - 1021\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{40000 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.75635, size = 167, normalized size = 1.44 \begin{align*} -\frac{1}{50000} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 69 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 199 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{3927}{20000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{6250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{3125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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