Optimal. Leaf size=94 \[ -\frac{2 (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{1}{5} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{33}{50} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]
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Rubi [A] time = 0.0222389, antiderivative size = 94, 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} \[ -\frac{2 (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{1}{5} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{33}{50} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]
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)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2}}{5 \sqrt{3+5 x}}-2 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{5 \sqrt{3+5 x}}-\frac{1}{5} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{33}{10} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{5 \sqrt{3+5 x}}-\frac{33}{50} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{5} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{363}{100} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{5 \sqrt{3+5 x}}-\frac{33}{50} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{5} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{363 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{50 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2}}{5 \sqrt{3+5 x}}-\frac{33}{50} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{5} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{50 \sqrt{10}}\\ \end{align*}
Mathematica [C] time = 0.0104912, size = 39, normalized size = 0.41 \[ -\frac{2}{77} \sqrt{\frac{2}{11}} (1-2 x)^{7/2} \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{5}{11} (1-2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.56164, size = 101, normalized size = 1.07 \begin{align*} -\frac{4 \, x^{3}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{17 \, x^{2}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{363}{1000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{223 \, x}{50 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{149}{50 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82743, size = 240, normalized size = 2.55 \begin{align*} \frac{363 \, \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 (20 \, x^{2} - 75 \, x - 149\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1000 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.2276, size = 230, normalized size = 2.45 \begin{align*} \begin{cases} \frac{4 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{5 \sqrt{10 x - 5}} - \frac{121 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{25 \sqrt{10 x - 5}} + \frac{121 i \sqrt{x + \frac{3}{5}}}{250 \sqrt{10 x - 5}} + \frac{363 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{500} + \frac{2662 i}{625 \sqrt{x + \frac{3}{5}} \sqrt{10 x - 5}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{500} - \frac{4 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{5 \sqrt{5 - 10 x}} + \frac{121 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{25 \sqrt{5 - 10 x}} - \frac{121 \sqrt{x + \frac{3}{5}}}{250 \sqrt{5 - 10 x}} - \frac{2662}{625 \sqrt{5 - 10 x} \sqrt{x + \frac{3}{5}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.72998, size = 150, normalized size = 1.6 \begin{align*} \frac{1}{1250} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 99 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{363}{500} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{625 \,{\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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