3.2023 \(\int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=41 \[ -\frac{3}{5} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

[Out]

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

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Rubi [A]  time = 0.0520136, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3}{5} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.97006, size = 37, normalized size = 0.9 \[ - \frac{3 \sqrt{- 2 x + 1}}{5} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{275} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0348391, size = 41, normalized size = 1. \[ -\frac{3}{5} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 29, normalized size = 0.7 \[ -{\frac{2\,\sqrt{55}}{275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{3}{5}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-3/5*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49397, size = 62, normalized size = 1.51 \[ \frac{1}{275} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3}{5} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244934, size = 65, normalized size = 1.59 \[ -\frac{1}{275} \, \sqrt{55}{\left (3 \, \sqrt{55} \sqrt{-2 \, x + 1} - \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.58038, size = 78, normalized size = 1.9 \[ - \frac{3 \sqrt{- 2 x + 1}}{5} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{5}{11} \end{cases}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-2*x + 1)/5 + 2*Piecewise((-sqrt(55)*acoth(sqrt(55)/(5*sqrt(-2*x + 1)))/
55, 1/(-2*x + 1) > 5/11), (-sqrt(55)*atanh(sqrt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-
2*x + 1) < 5/11))/5

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GIAC/XCAS [A]  time = 0.214903, size = 66, normalized size = 1.61 \[ \frac{1}{275} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3}{5} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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