Optimal. Leaf size=52 \[ -\frac{2 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}-\frac{2}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
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Rubi [A] time = 0.0446696, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}-\frac{2}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 2*x]/(3 + 5*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 5.21063, size = 46, normalized size = 0.88 \[ - \frac{2 \sqrt{- 2 x + 1}}{5 \sqrt{5 x + 3}} - \frac{2 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(1/2)/(3+5*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0868619, size = 49, normalized size = 0.94 \[ \frac{2}{25} \left (\sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{5 \sqrt{1-2 x}}{\sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 2*x]/(3 + 5*x)^(3/2),x]
[Out]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int{1\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(1/2)/(3+5*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.50837, size = 49, normalized size = 0.94 \[ -\frac{1}{25} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{5 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217918, size = 93, normalized size = 1.79 \[ -\frac{\sqrt{5}{\left (\sqrt{2}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{25 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.70376, size = 153, normalized size = 2.94 \[ \begin{cases} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{25} - \frac{\sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{25} - \frac{\sqrt{10} i \log{\left (x + \frac{3}{5} \right )}}{25} - \frac{2 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{25} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{25} - \frac{\sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{25} + \frac{2 \sqrt{10} i \log{\left (\sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} + 1 \right )}}{25} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(1/2)/(3+5*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227145, size = 112, normalized size = 2.15 \[ -\frac{1}{50} \, \sqrt{5}{\left (4 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{2} \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^(3/2),x, algorithm="giac")
[Out]