3.2036 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0419926, 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 \[ -\frac{\sqrt{1-2 x}}{11 (5 x+3)}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.51915, size = 39, normalized size = 0.81 \[ - \frac{\sqrt{- 2 x + 1}}{11 \left (5 x + 3\right )} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{605} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)/(11*(5*x + 3)) - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/60
5

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Mathematica [A]  time = 0.0638548, size = 46, normalized size = 0.96 \[ -\frac{\sqrt{1-2 x}}{55 x+33}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]/(33 + 55*x)) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(11*Sqrt[55
])

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Maple [A]  time = 0.01, size = 36, normalized size = 0.8 \[{\frac{2}{-66-110\,x}\sqrt{1-2\,x}}-{\frac{2\,\sqrt{55}}{605}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(1-2*x)^(1/2)/(-6-10*x)-2/605*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50446, size = 72, normalized size = 1.5 \[ \frac{1}{605} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{\sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/605*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 1/11*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.230038, size = 80, normalized size = 1.67 \[ \frac{\sqrt{55}{\left ({\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55} \sqrt{-2 \, x + 1}\right )}}{605 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/605*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 = 3.13578, size = 173, normalized size = 3.6 \[ \begin{cases} - \frac{2 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{605} + \frac{\sqrt{2}}{55 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{\sqrt{2}}{50 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{2 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{605} - \frac{\sqrt{2} i}{55 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{\sqrt{2} i}{50 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/605 + sqrt(2)/(55*sqr
t(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - sqrt(2)/(50*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)))/605 - sqrt(2)*I/(55*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/
5)) + sqrt(2)*I/(50*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), True))

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GIAC/XCAS [A]  time = 0.210147, size = 76, normalized size = 1.58 \[ \frac{1}{605} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{\sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/605*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) - 1/11*sqrt(-2*x + 1)/(5*x + 3)