Optimal. Leaf size=96 \[ -\frac{2 (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
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Rubi [A] time = 0.0233207, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {47, 50, 54, 216} \[ -\frac{2 (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{2}{3} \int \frac{(1-2 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{3+5 x}}+\frac{4}{5} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{3+5 x}}+\frac{4}{25} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{22}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{3+5 x}}+\frac{4}{25} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{44 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{3+5 x}}+\frac{4}{25} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{22}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0105477, size = 39, normalized size = 0.41 \[ -\frac{4}{847} \sqrt{\frac{2}{11}} (1-2 x)^{7/2} \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{5}{11} (1-2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.51408, size = 174, normalized size = 1.81 \begin{align*} \frac{11}{125} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{30 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{150 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{77 \, \sqrt{-10 \, x^{2} - x + 3}}{75 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69368, size = 285, normalized size = 2.97 \begin{align*} -\frac{33 \, \sqrt{5} \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 10 \,{\left (30 \, x^{2} + 190 \, x + 79\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{375 \,{\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 [C] time = 15.6286, size = 257, normalized size = 2.68 \begin{align*} \begin{cases} \frac{4 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{125} + \frac{308 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875} - \frac{242 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{9375 \left (x + \frac{3}{5}\right )} + \frac{11 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} + \frac{11 \sqrt{10} i \log{\left (x + \frac{3}{5} \right )}}{125} + \frac{22 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\\frac{4 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{125} + \frac{308 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875} - \frac{242 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{9375 \left (x + \frac{3}{5}\right )} + \frac{11 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} - \frac{22 \sqrt{10} i \log{\left (\sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} + 1 \right )}}{125} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.2304, size = 220, normalized size = 2.29 \begin{align*} -\frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{30000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{4}{625} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{22}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{99 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2500 \, \sqrt{5 \, x + 3}} - \frac{11 \,{\left (\frac{27 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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