3.2046 \(\int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{69 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{110 (5 x+3)^2}-\frac{69 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

[Out]

-Sqrt[1 - 2*x]/(110*(3 + 5*x)^2) - (69*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (69*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

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Rubi [A]  time = 0.0714474, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{69 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{110 (5 x+3)^2}-\frac{69 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - 2*x]/(110*(3 + 5*x)^2) - (69*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (69*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

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Rubi in Sympy [A]  time = 6.99129, size = 58, normalized size = 0.85 \[ - \frac{69 \sqrt{- 2 x + 1}}{1210 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1}}{110 \left (5 x + 3\right )^{2}} - \frac{69 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{33275} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-69*sqrt(-2*x + 1)/(1210*(5*x + 3)) - sqrt(-2*x + 1)/(110*(5*x + 3)**2) - 69*sqr
t(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/33275

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Mathematica [A]  time = 0.0982949, size = 53, normalized size = 0.78 \[ -\frac{\sqrt{1-2 x} (345 x+218)}{1210 (5 x+3)^2}-\frac{69 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(218 + 345*x))/(1210*(3 + 5*x)^2) - (69*ArcTanh[Sqrt[5/11]*Sqrt[
1 - 2*x]])/(605*Sqrt[55])

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Maple [A]  time = 0.013, size = 48, normalized size = 0.7 \[ -100\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{69\, \left ( 1-2\,x \right ) ^{3/2}}{12100}}+{\frac{71\,\sqrt{1-2\,x}}{5500}} \right ) }-{\frac{69\,\sqrt{55}}{33275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-100*(-69/12100*(1-2*x)^(3/2)+71/5500*(1-2*x)^(1/2))/(-6-10*x)^2-69/33275*arctan
h(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50782, size = 100, normalized size = 1.47 \[ \frac{69}{66550} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{345 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 781 \, \sqrt{-2 \, x + 1}}{605 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

69/66550*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 1/605*(345*(-2*x + 1)^(3/2) - 781*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x
+ 11)

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Fricas [A]  time = 0.226754, size = 100, normalized size = 1.47 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (345 \, x + 218\right )} \sqrt{-2 \, x + 1} - 69 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{66550 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/66550*sqrt(55)*(sqrt(55)*(345*x + 218)*sqrt(-2*x + 1) - 69*(25*x^2 + 30*x + 9
)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2 + 30*x + 9)

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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)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.232515, size = 92, normalized size = 1.35 \[ \frac{69}{66550} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{345 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 781 \, \sqrt{-2 \, x + 1}}{2420 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

69/66550*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 1/2420*(345*(-2*x + 1)^(3/2) - 781*sqrt(-2*x + 1))/(5*x + 3)^2