3.2092 \(\int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{7}{11 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]

[Out]

7/(11*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(11*Sqrt[55])

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Rubi [A]  time = 0.0515368, 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 \[ \frac{7}{11 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

7/(11*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(11*Sqrt[55])

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Rubi in Sympy [A]  time = 5.37215, size = 36, normalized size = 0.88 \[ - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{605} + \frac{7}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/605 + 7/(11*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0464046, size = 41, normalized size = 1. \[ \frac{7}{11 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{11 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

7/(11*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(11*Sqrt[55])

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Maple [A]  time = 0.011, size = 29, normalized size = 0.7 \[ -{\frac{2\,\sqrt{55}}{605}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{7}{11}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/605*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/11/(1-2*x)^(1/2)

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Maxima [A]  time = 1.49624, size = 62, normalized size = 1.51 \[ \frac{1}{605} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7}{11 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/605*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 7/11/sqrt(-2*x + 1)

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Fricas [A]  time = 0.238801, size = 73, normalized size = 1.78 \[ \frac{\sqrt{55}{\left (\sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 7 \, \sqrt{55}\right )}}{605 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/605*sqrt(55)*(sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x
+ 3)) + 7*sqrt(55))/sqrt(-2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 x + 2}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)/((-2*x + 1)**(3/2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.216894, size = 66, normalized size = 1.61 \[ \frac{1}{605} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{7}{11 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)/((5*x + 3)*(-2*x + 1)^(3/2)),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))) + 7/11/sqrt(-2*x + 1)