Optimal. Leaf size=48 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}}-\frac{\sqrt{1-2 x}}{5 (5 x+3)} \]
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Rubi [A] time = 0.0373359, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}}-\frac{\sqrt{1-2 x}}{5 (5 x+3)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 2*x]/(3 + 5*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 4.8094, size = 37, normalized size = 0.77 \[ - \frac{\sqrt{- 2 x + 1}}{5 \left (5 x + 3\right )} + \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{275} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(1/2)/(3+5*x)**2,x)
[Out]
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Mathematica [A] time = 0.0592442, size = 48, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}}-\frac{\sqrt{1-2 x}}{5 (5 x+3)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 2*x]/(3 + 5*x)^2,x]
[Out]
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Maple [A] time = 0.012, size = 36, normalized size = 0.8 \[{\frac{2}{25}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{2\,\sqrt{55}}{275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(1/2)/(3+5*x)^2,x)
[Out]
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Maxima [A] time = 1.4925, size = 72, normalized size = 1.5 \[ -\frac{1}{275} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{\sqrt{-2 \, x + 1}}{5 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210272, size = 80, normalized size = 1.67 \[ \frac{\sqrt{55}{\left ({\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} - 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55} \sqrt{-2 \, x + 1}\right )}}{275 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.52035, size = 177, normalized size = 3.69 \[ \begin{cases} \frac{2 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{275} + \frac{\sqrt{2}}{25 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{11 \sqrt{2}}{250 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{2 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{275} - \frac{\sqrt{2} i}{25 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{11 \sqrt{2} i}{250 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(1/2)/(3+5*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.212071, size = 76, normalized size = 1.58 \[ -\frac{1}{275} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{\sqrt{-2 \, x + 1}}{5 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^2,x, algorithm="giac")
[Out]