3.2045 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{27 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{550 (5 x+3)^2}-\frac{2313 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

[Out]

-Sqrt[1 - 2*x]/(550*(3 + 5*x)^2) - (27*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (2313*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi [A]  time = 0.0899744, antiderivative size = 68, 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 \[ -\frac{27 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{550 (5 x+3)^2}-\frac{2313 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - 2*x]/(550*(3 + 5*x)^2) - (27*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (2313*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi in Sympy [A]  time = 8.41899, size = 58, normalized size = 0.85 \[ - \frac{27 \sqrt{- 2 x + 1}}{1210 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1}}{550 \left (5 x + 3\right )^{2}} - \frac{2313 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{166375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*sqrt(-2*x + 1)/(1210*(5*x + 3)) - sqrt(-2*x + 1)/(550*(5*x + 3)**2) - 2313*s
qrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/166375

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Mathematica [A]  time = 0.0887703, size = 53, normalized size = 0.78 \[ \frac{-\frac{55 \sqrt{1-2 x} (675 x+416)}{(5 x+3)^2}-4626 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{332750} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(416 + 675*x))/(3 + 5*x)^2 - 4626*Sqrt[55]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/332750

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Maple [A]  time = 0.016, size = 48, normalized size = 0.7 \[ 50\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ({\frac{27\, \left ( 1-2\,x \right ) ^{3/2}}{6050}}-{\frac{137\,\sqrt{1-2\,x}}{13750}} \right ) }-{\frac{2313\,\sqrt{55}}{166375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50*(27/6050*(1-2*x)^(3/2)-137/13750*(1-2*x)^(1/2))/(-6-10*x)^2-2313/166375*arcta
nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49961, size = 100, normalized size = 1.47 \[ \frac{2313}{332750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{675 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1507 \, \sqrt{-2 \, x + 1}}{3025 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2313/332750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 1/3025*(675*(-2*x + 1)^(3/2) - 1507*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 2
20*x + 11)

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Fricas [A]  time = 0.237017, size = 100, normalized size = 1.47 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (675 \, x + 416\right )} \sqrt{-2 \, x + 1} - 2313 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{332750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/332750*sqrt(55)*(sqrt(55)*(675*x + 416)*sqrt(-2*x + 1) - 2313*(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)**2/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.22575, size = 92, normalized size = 1.35 \[ \frac{2313}{332750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{675 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1507 \, \sqrt{-2 \, x + 1}}{12100 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2313/332750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) + 1/12100*(675*(-2*x + 1)^(3/2) - 1507*sqrt(-2*x + 1))/(5*x + 3)
^2