3.2034 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

Optimal. Leaf size=61 \[ -\frac{9}{25} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{275 (5 x+3)}-\frac{134 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

[Out]

(-9*Sqrt[1 - 2*x])/25 - Sqrt[1 - 2*x]/(275*(3 + 5*x)) - (134*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi [A]  time = 0.0798838, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{9}{25} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{275 (5 x+3)}-\frac{134 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-9*Sqrt[1 - 2*x])/25 - Sqrt[1 - 2*x]/(275*(3 + 5*x)) - (134*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi in Sympy [A]  time = 7.73213, size = 51, normalized size = 0.84 \[ - \frac{9 \sqrt{- 2 x + 1}}{25} - \frac{\sqrt{- 2 x + 1}}{275 \left (5 x + 3\right )} - \frac{134 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*sqrt(-2*x + 1)/25 - sqrt(-2*x + 1)/(275*(5*x + 3)) - 134*sqrt(55)*atanh(sqrt(
55)*sqrt(-2*x + 1)/11)/15125

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Mathematica [A]  time = 0.0941835, size = 53, normalized size = 0.87 \[ -\frac{\sqrt{1-2 x} (495 x+298)}{275 (5 x+3)}-\frac{134 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(298 + 495*x))/(275*(3 + 5*x)) - (134*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/(275*Sqrt[55])

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Maple [A]  time = 0.016, size = 45, normalized size = 0.7 \[ -{\frac{9}{25}\sqrt{1-2\,x}}+{\frac{2}{1375}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{134\,\sqrt{55}}{15125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/25*(1-2*x)^(1/2)+2/1375*(1-2*x)^(1/2)/(-6/5-2*x)-134/15125*arctanh(1/11*55^(1
/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.4988, size = 84, normalized size = 1.38 \[ \frac{67}{15125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{275 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

67/15125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 9/25*sqrt(-2*x + 1) - 1/275*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.238598, size = 86, normalized size = 1.41 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (495 \, x + 298\right )} \sqrt{-2 \, x + 1} - 67 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{15125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15125*sqrt(55)*(sqrt(55)*(495*x + 298)*sqrt(-2*x + 1) - 67*(5*x + 3)*log((sqr
t(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.215002, size = 88, normalized size = 1.44 \[ \frac{67}{15125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{9}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{275 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

67/15125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 9/25*sqrt(-2*x + 1) - 1/275*sqrt(-2*x + 1)/(5*x + 3)