3.1815 \(\int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\)

Optimal. Leaf size=43 \[ \frac{2}{5} \sqrt{1-2 x}-\frac{2}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/5 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi [A]  time = 0.0375814, antiderivative size = 43, 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 \[ \frac{2}{5} \sqrt{1-2 x}-\frac{2}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]/(3 + 5*x),x]

[Out]

(2*Sqrt[1 - 2*x])/5 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi in Sympy [A]  time = 4.54336, size = 36, normalized size = 0.84 \[ \frac{2 \sqrt{- 2 x + 1}}{5} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(1/2)/(3+5*x),x)

[Out]

2*sqrt(-2*x + 1)/5 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/25

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Mathematica [A]  time = 0.0258767, size = 41, normalized size = 0.95 \[ \frac{2}{25} \left (5 \sqrt{1-2 x}-\sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 2*x]/(3 + 5*x),x]

[Out]

(2*(5*Sqrt[1 - 2*x] - Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/25

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Maple [A]  time = 0.008, size = 29, normalized size = 0.7 \[ -{\frac{2\,\sqrt{55}}{25}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{5}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(1/2)/(3+5*x),x)

[Out]

-2/25*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/5*(1-2*x)^(1/2)

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Maxima [A]  time = 1.50408, size = 62, normalized size = 1.44 \[ \frac{1}{25} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{5} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/(5*x + 3),x, algorithm="maxima")

[Out]

1/25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 2/5*sqrt(-2*x + 1)

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Fricas [A]  time = 0.216477, size = 72, normalized size = 1.67 \[ \frac{1}{25} \, \sqrt{5}{\left (\sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 2 \, \sqrt{5} \sqrt{-2 \, x + 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/(5*x + 3),x, algorithm="fricas")

[Out]

1/25*sqrt(5)*(sqrt(11)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x
+ 3)) + 2*sqrt(5)*sqrt(-2*x + 1))

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Sympy [A]  time = 4.57064, size = 107, normalized size = 2.49 \[ \begin{cases} \frac{2 \sqrt{5} i \sqrt{10 x - 5}}{25} + \frac{2 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{25} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{2 \sqrt{5} \sqrt{- 10 x + 5}}{25} + \frac{\sqrt{55} \log{\left (x + \frac{3}{5} \right )}}{25} - \frac{2 \sqrt{55} \log{\left (\sqrt{- \frac{10 x}{11} + \frac{5}{11}} + 1 \right )}}{25} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(1/2)/(3+5*x),x)

[Out]

Piecewise((2*sqrt(5)*I*sqrt(10*x - 5)/25 + 2*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(
x + 3/5)))/25, 10*Abs(x + 3/5)/11 > 1), (2*sqrt(5)*sqrt(-10*x + 5)/25 + sqrt(55)
*log(x + 3/5)/25 - 2*sqrt(55)*log(sqrt(-10*x/11 + 5/11) + 1)/25, True))

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GIAC/XCAS [A]  time = 0.215221, size = 66, normalized size = 1.53 \[ \frac{1}{25} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2}{5} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/(5*x + 3),x, algorithm="giac")

[Out]

1/25*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) + 2/5*sqrt(-2*x + 1)