Optimal. Leaf size=68 \[ -\frac{3 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{\sqrt{1-2 x}}{22 (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]
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Rubi [A] time = 0.0598688, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{3 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{\sqrt{1-2 x}}{22 (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]
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Rubi in Sympy [A] time = 6.09774, size = 58, normalized size = 0.85 \[ - \frac{3 \sqrt{- 2 x + 1}}{242 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1}}{22 \left (5 x + 3\right )^{2}} - \frac{3 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{6655} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(3+5*x)**3/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0775396, size = 53, normalized size = 0.78 \[ -\frac{5 \sqrt{1-2 x} (3 x+4)}{242 (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 - 2*x]*(3 + 5*x)^3),x]
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Maple [A] time = 0.009, size = 52, normalized size = 0.8 \[ -{\frac{2}{11\, \left ( -6-10\,x \right ) ^{2}}\sqrt{1-2\,x}}+{\frac{3}{-726-1210\,x}\sqrt{1-2\,x}}-{\frac{3\,\sqrt{55}}{6655}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(3+5*x)^3/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.51663, size = 100, normalized size = 1.47 \[ \frac{3}{13310} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{5 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}}{121 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^3*sqrt(-2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21916, size = 101, normalized size = 1.49 \[ -\frac{\sqrt{55}{\left (5 \, \sqrt{55}{\left (3 \, x + 4\right )} \sqrt{-2 \, x + 1} - 3 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{13310 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^3*sqrt(-2*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.10855, size = 233, normalized size = 3.43 \[ \begin{cases} - \frac{3 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{6655} + \frac{3 \sqrt{2}}{1210 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{\sqrt{2}}{1100 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{\sqrt{2}}{500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{3 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{6655} - \frac{3 \sqrt{2} i}{1210 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{\sqrt{2} i}{1100 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} i}{500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(3+5*x)**3/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.218386, size = 92, normalized size = 1.35 \[ \frac{3}{13310} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{5 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^3*sqrt(-2*x + 1)),x, algorithm="giac")
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