3.1826 \(\int \frac{\sqrt{1-2 x}}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}}-\frac{\sqrt{1-2 x}}{5 (5 x+3)} \]

[Out]

-Sqrt[1 - 2*x]/(5*(3 + 5*x)) + (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]
)

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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]

-Sqrt[1 - 2*x]/(5*(3 + 5*x)) + (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]
)

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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]

-sqrt(-2*x + 1)/(5*(5*x + 3)) + 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/275

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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]

-Sqrt[1 - 2*x]/(5*(3 + 5*x)) + (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]
)

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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]

2/25*(1-2*x)^(1/2)/(-6/5-2*x)+2/275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2
)

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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]

-1/275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 1/5*sqrt(-2*x + 1)/(5*x + 3)

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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]

1/275*sqrt(55)*((5*x + 3)*log((sqrt(55)*(5*x - 8) - 55*sqrt(-2*x + 1))/(5*x + 3)
) - sqrt(55)*sqrt(-2*x + 1))/(5*x + 3)

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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]

Piecewise((2*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/275 + sqrt(2)/(25*sqrt
(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - 11*sqrt(2)/(250*sqrt(-1 + 11/(10*(x +
3/5)))*(x + 3/5)**(3/2)), 11*Abs(1/(x + 3/5))/10 > 1), (-2*sqrt(55)*I*asin(sqrt(
110)/(10*sqrt(x + 3/5)))/275 - sqrt(2)*I/(25*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x
+ 3/5)) + 11*sqrt(2)*I/(250*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), True)
)

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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]

-1/275*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 1/5*sqrt(-2*x + 1)/(5*x + 3)