3.2024 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0265375, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.06754, size = 26, normalized size = 1.04 \[ - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0112023, size = 25, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 19, normalized size = 0.8 \[ -{\frac{2\,\sqrt{55}}{55}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3+5*x)/(1-2*x)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.4794, size = 49, normalized size = 1.96 \[ \frac{1}{55} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.22675, size = 45, normalized size = 1.8 \[ \frac{1}{55} \, \sqrt{55} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.74909, size = 63, normalized size = 2.52 \[ \begin{cases} - \frac{2 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{55} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{2 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{55} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224098, size = 54, normalized size = 2.16 \[ -\frac{1}{55} \, \sqrt{55}{\rm ln}\left (\frac{1}{5} \, \sqrt{55} + \sqrt{-2 \, x + 1}\right ) + \frac{1}{55} \, \sqrt{55}{\rm ln}\left ({\left | -\frac{1}{5} \, \sqrt{55} + \sqrt{-2 \, x + 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/55*sqrt(55)*ln(1/5*sqrt(55) + sqrt(-2*x + 1)) + 1/55*sqrt(55)*ln(abs(-1/5*sqr
t(55) + sqrt(-2*x + 1)))