Optimal. Leaf size=68 \[ \frac{\sqrt{1-2 x}}{110 (5 x+3)}-\frac{\sqrt{1-2 x}}{10 (5 x+3)^2}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]
[Out]
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Rubi [A] time = 0.0535309, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{\sqrt{1-2 x}}{110 (5 x+3)}-\frac{\sqrt{1-2 x}}{10 (5 x+3)^2}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 2*x]/(3 + 5*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 6.6489, size = 53, normalized size = 0.78 \[ \frac{\sqrt{- 2 x + 1}}{110 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1}}{10 \left (5 x + 3\right )^{2}} + \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3025} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(1/2)/(3+5*x)**3,x)
[Out]
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Mathematica [A] time = 0.0789328, size = 53, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} (5 x-8)}{110 (5 x+3)^2}+\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 2*x]/(3 + 5*x)^3,x]
[Out]
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Maple [A] time = 0.014, size = 48, normalized size = 0.7 \[ 200\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{3/2}}{2200}}-{\frac{\sqrt{1-2\,x}}{1000}} \right ) }+{\frac{\sqrt{55}}{3025}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(1/2)/(3+5*x)^3,x)
[Out]
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Maxima [A] time = 1.49177, size = 100, normalized size = 1.47 \[ -\frac{1}{6050} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 11 \, \sqrt{-2 \, x + 1}}{55 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215783, size = 99, normalized size = 1.46 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (5 \, x - 8\right )} \sqrt{-2 \, x + 1} +{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} - 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{6050 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.32716, size = 233, normalized size = 3.43 \[ \begin{cases} \frac{\sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{3025} - \frac{\sqrt{2}}{550 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{3 \sqrt{2}}{500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{11 \sqrt{2}}{2500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{\sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{3025} + \frac{\sqrt{2} i}{550 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{3 \sqrt{2} i}{500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{11 \sqrt{2} i}{2500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(1/2)/(3+5*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.212118, size = 92, normalized size = 1.35 \[ -\frac{1}{6050} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 11 \, \sqrt{-2 \, x + 1}}{220 \,{\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/(5*x + 3)^3,x, algorithm="giac")
[Out]