3.1 \(\int \frac{1}{2 \sqrt{3} b^{3/2}-9 b x+9 x^3} \, dx\)

Optimal. Leaf size=77 \[ \frac{1}{3 \sqrt{3} \sqrt{b} \left (\sqrt{3} \sqrt{b}-3 x\right )}-\frac{\log \left (\sqrt{b}-\sqrt{3} x\right )}{27 b}+\frac{\log \left (2 \sqrt{b}+\sqrt{3} x\right )}{27 b} \]

[Out]

1/(3*Sqrt[3]*Sqrt[b]*(Sqrt[3]*Sqrt[b] - 3*x)) - Log[Sqrt[b] - Sqrt[3]*x]/(27*b)
+ Log[2*Sqrt[b] + Sqrt[3]*x]/(27*b)

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Rubi [A]  time = 0.130636, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{3 \sqrt{3} \sqrt{b} \left (\sqrt{3} \sqrt{b}-3 x\right )}-\frac{\log \left (\sqrt{b}-\sqrt{3} x\right )}{27 b}+\frac{\log \left (2 \sqrt{b}+\sqrt{3} x\right )}{27 b} \]

Antiderivative was successfully verified.

[In]  Int[(2*Sqrt[3]*b^(3/2) - 9*b*x + 9*x^3)^(-1),x]

[Out]

1/(3*Sqrt[3]*Sqrt[b]*(Sqrt[3]*Sqrt[b] - 3*x)) - Log[Sqrt[b] - Sqrt[3]*x]/(27*b)
+ Log[2*Sqrt[b] + Sqrt[3]*x]/(27*b)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-9*b*x+9*x**3+2*b**(3/2)*3**(1/2)),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0697358, size = 143, normalized size = 1.86 \[ -\frac{\left (3 x-\sqrt{3} \sqrt{b}\right ) \left (2 \sqrt{3} \sqrt{b}+3 x\right ) \left (\left (3 x-\sqrt{3} \sqrt{b}\right ) \log \left (3 x-\sqrt{3} \sqrt{b}\right )+\left (\sqrt{3} \sqrt{b}-3 x\right ) \log \left (2 \sqrt{3} \sqrt{b}+3 x\right )+3 \sqrt{3} \sqrt{b}\right )}{81 b \left (2 \sqrt{3} b^{3/2}-9 b x+9 x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(2*Sqrt[3]*b^(3/2) - 9*b*x + 9*x^3)^(-1),x]

[Out]

-((-(Sqrt[3]*Sqrt[b]) + 3*x)*(2*Sqrt[3]*Sqrt[b] + 3*x)*(3*Sqrt[3]*Sqrt[b] + (-(S
qrt[3]*Sqrt[b]) + 3*x)*Log[-(Sqrt[3]*Sqrt[b]) + 3*x] + (Sqrt[3]*Sqrt[b] - 3*x)*L
og[2*Sqrt[3]*Sqrt[b] + 3*x]))/(81*b*(2*Sqrt[3]*b^(3/2) - 9*b*x + 9*x^3))

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Maple [C]  time = 0.009, size = 43, normalized size = 0.6 \[{\frac{1}{9}\sum _{{\it \_R}={\it RootOf} \left ( -9\,{\it \_Z}\,b+9\,{{\it \_Z}}^{3}+2\,{b}^{3/2}\sqrt{3} \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}-b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-9*b*x+9*x^3+2*b^(3/2)*3^(1/2)),x)

[Out]

1/9*sum(1/(3*_R^2-b)*ln(x-_R),_R=RootOf(-9*_Z*b+9*_Z^3+2*b^(3/2)*3^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{9 \, x^{3} + 2 \, \sqrt{3} b^{\frac{3}{2}} - 9 \, b x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(9*x^3 + 2*sqrt(3)*b^(3/2) - 9*b*x),x, algorithm="maxima")

[Out]

integrate(1/(9*x^3 + 2*sqrt(3)*b^(3/2) - 9*b*x), x)

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Fricas [A]  time = 0.286915, size = 105, normalized size = 1.36 \[ \frac{{\left (\sqrt{3} b - 3 \, \sqrt{b} x\right )} \log \left (\sqrt{3} \sqrt{b} x + 2 \, b\right ) -{\left (\sqrt{3} b - 3 \, \sqrt{b} x\right )} \log \left (\sqrt{3} \sqrt{b} x - b\right ) + 3 \, \sqrt{3} b}{27 \,{\left (\sqrt{3} b^{2} - 3 \, b^{\frac{3}{2}} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(9*x^3 + 2*sqrt(3)*b^(3/2) - 9*b*x),x, algorithm="fricas")

[Out]

1/27*((sqrt(3)*b - 3*sqrt(b)*x)*log(sqrt(3)*sqrt(b)*x + 2*b) - (sqrt(3)*b - 3*sq
rt(b)*x)*log(sqrt(3)*sqrt(b)*x - b) + 3*sqrt(3)*b)/(sqrt(3)*b^2 - 3*b^(3/2)*x)

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Sympy [A]  time = 1.7554, size = 60, normalized size = 0.78 \[ - \frac{3 \sqrt{3}}{81 \sqrt{b} x - 27 \sqrt{3} b} + \frac{- \frac{\log{\left (- \frac{\sqrt{3} \sqrt{b}}{3} + x \right )}}{27} + \frac{\log{\left (\frac{2 \sqrt{3} \sqrt{b}}{3} + x \right )}}{27}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-9*b*x+9*x**3+2*b**(3/2)*3**(1/2)),x)

[Out]

-3*sqrt(3)/(81*sqrt(b)*x - 27*sqrt(3)*b) + (-log(-sqrt(3)*sqrt(b)/3 + x)/27 + lo
g(2*sqrt(3)*sqrt(b)/3 + x)/27)/b

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GIAC/XCAS [A]  time = 0.280025, size = 73, normalized size = 0.95 \[ \frac{{\rm ln}\left ({\left | \sqrt{3} x + 2 \, \sqrt{b} \right |}\right )}{27 \, b} - \frac{{\rm ln}\left ({\left | -\sqrt{3} x + \sqrt{b} \right |}\right )}{27 \, b} - \frac{1}{9 \,{\left (\sqrt{3} x - \sqrt{b}\right )} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(9*x^3 + 2*sqrt(3)*b^(3/2) - 9*b*x),x, algorithm="giac")

[Out]

1/27*ln(abs(sqrt(3)*x + 2*sqrt(b)))/b - 1/27*ln(abs(-sqrt(3)*x + sqrt(b)))/b - 1
/9/((sqrt(3)*x - sqrt(b))*sqrt(b))