3.2091 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=54 \[ \frac{9}{10} \sqrt{1-2 x}+\frac{49}{22 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

[Out]

49/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x])/10 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(55*Sqrt[55])

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Rubi [A]  time = 0.0835943, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{9}{10} \sqrt{1-2 x}+\frac{49}{22 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

49/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x])/10 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(55*Sqrt[55])

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Rubi in Sympy [A]  time = 10.1355, size = 48, normalized size = 0.89 \[ \frac{9 \sqrt{- 2 x + 1}}{10} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3025} + \frac{49}{22 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(-2*x + 1)/10 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/3025 + 49/(22
*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0929973, size = 46, normalized size = 0.85 \[ \frac{172-99 x}{55 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(172 - 99*x)/(55*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(55*Sqrt
[55])

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Maple [A]  time = 0.013, size = 38, normalized size = 0.7 \[ -{\frac{2\,\sqrt{55}}{3025}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{49}{22}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{9}{10}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3025*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+49/22/(1-2*x)^(1/2)+9/10*(
1-2*x)^(1/2)

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Maxima [A]  time = 1.51252, size = 74, normalized size = 1.37 \[ \frac{1}{3025} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{9}{10} \, \sqrt{-2 \, x + 1} + \frac{49}{22 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3025*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) + 9/10*sqrt(-2*x + 1) + 49/22/sqrt(-2*x + 1)

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Fricas [A]  time = 0.242635, size = 80, normalized size = 1.48 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (99 \, x - 172\right )} - \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{3025 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3025*sqrt(55)*(sqrt(55)*(99*x - 172) - sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8)
+ 55*sqrt(-2*x + 1))/(5*x + 3)))/sqrt(-2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246726, size = 78, normalized size = 1.44 \[ \frac{1}{3025} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{9}{10} \, \sqrt{-2 \, x + 1} + \frac{49}{22 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3025*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + 9/10*sqrt(-2*x + 1) + 49/22/sqrt(-2*x + 1)