3.2035 \(\int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{\sqrt{1-2 x}}{55 (5 x+3)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

[Out]

-Sqrt[1 - 2*x]/(55*(3 + 5*x)) - (68*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(55*Sqrt[
55])

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Rubi [A]  time = 0.0518987, antiderivative size = 48, 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 \[ -\frac{\sqrt{1-2 x}}{55 (5 x+3)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - 2*x]/(55*(3 + 5*x)) - (68*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(55*Sqrt[
55])

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Rubi in Sympy [A]  time = 5.35974, size = 39, normalized size = 0.81 \[ - \frac{\sqrt{- 2 x + 1}}{55 \left (5 x + 3\right )} - \frac{68 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3025} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)/(55*(5*x + 3)) - 68*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/3
025

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Mathematica [A]  time = 0.0723926, size = 48, normalized size = 1. \[ -\frac{\sqrt{1-2 x}}{55 (5 x+3)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - 2*x]/(55*(3 + 5*x)) - (68*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(55*Sqrt[
55])

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Maple [A]  time = 0.015, size = 36, normalized size = 0.8 \[{\frac{2}{275}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{68\,\sqrt{55}}{3025}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/275*(1-2*x)^(1/2)/(-6/5-2*x)-68/3025*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(
1/2)

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Maxima [A]  time = 1.49468, size = 72, normalized size = 1.5 \[ \frac{34}{3025} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{\sqrt{-2 \, x + 1}}{55 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34/3025*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 1/55*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.227967, size = 81, normalized size = 1.69 \[ \frac{\sqrt{55}{\left (34 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55} \sqrt{-2 \, x + 1}\right )}}{3025 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3025*sqrt(55)*(34*(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 [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)/(3+5*x)**2/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.21872, size = 76, normalized size = 1.58 \[ \frac{34}{3025} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{\sqrt{-2 \, x + 1}}{55 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34/3025*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 1/55*sqrt(-2*x + 1)/(5*x + 3)