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 = 0.13389, 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, Rules used = {2063, 44} \[ \frac{1}{3 \sqrt{3} \sqrt{b} \left (\sqrt{3} \sqrt{b}-3 x\right )}-\frac{\log \left (\sqrt{b}-\sqrt{3} x\right )}{27 b}+\frac{\log \left (2 \sqrt{b}+\sqrt{3} x\right )}{27 b} \]

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)

Rule 2063

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[1/(3^(3*p)*a^(2*p)), Int[(3*a - b*x)^p*(3*a +
2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{2 \sqrt{3} b^{3/2}-9 b x+9 x^3} \, dx &=\left (324 b^3\right ) \int \frac{1}{\left (6 \sqrt{3} b^{3/2}-18 b x\right )^2 \left (6 \sqrt{3} b^{3/2}+9 b x\right )} \, dx\\ &=\left (324 b^3\right ) \int \left (\frac{1}{324 \sqrt{3} b^{7/2} \left (\sqrt{3} \sqrt{b}-3 x\right )^2}+\frac{1}{2916 b^4 \left (\sqrt{3} \sqrt{b}-3 x\right )}+\frac{1}{2916 b^4 \left (2 \sqrt{3} \sqrt{b}+3 x\right )}\right ) \, dx\\ &=\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}\\ \end{align*}

Mathematica [A]  time = 0.0484474, 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] + (-(Sqrt[3]*Sqrt[b]) + 3*x)*Log[-(S
qrt[3]*Sqrt[b]) + 3*x] + (Sqrt[3]*Sqrt[b] - 3*x)*Log[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 = 0.007, size = 43, normalized size = 0.6 \begin{align*}{\frac{1}{9}\sum _{{\it \_R}={\it RootOf} \left ( -9\,b{\it \_Z}+9\,{{\it \_Z}}^{3}+2\,{b}^{3/2}\sqrt{3} \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}-b}}} \end{align*}

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*b*_Z+9*_Z^3+2*b^(3/2)*3^(1/2)))

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

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, 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 = 1.38785, size = 184, normalized size = 2.39 \begin{align*} -\frac{3 \, \sqrt{3} \sqrt{b} x -{\left (3 \, x^{2} - b\right )} \log \left (2 \, \sqrt{3} \sqrt{b} + 3 \, x\right ) +{\left (3 \, x^{2} - b\right )} \log \left (-\sqrt{3} \sqrt{b} + 3 \, x\right ) + 3 \, b}{27 \,{\left (3 \, b x^{2} - b^{2}\right )}} \end{align*}

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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.443516, size = 60, normalized size = 0.78 \begin{align*} - \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} \end{align*}

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 + log(2*sqrt(3)*sqrt(b)/3 + x)/27)
/b

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Giac [A]  time = 1.09659, size = 73, normalized size = 0.95 \begin{align*} \frac{\log \left ({\left | \sqrt{3} x + 2 \, \sqrt{b} \right |}\right )}{27 \, b} - \frac{\log \left ({\left | -\sqrt{3} x + \sqrt{b} \right |}\right )}{27 \, b} - \frac{1}{9 \,{\left (\sqrt{3} x - \sqrt{b}\right )} \sqrt{b}} \end{align*}

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, algorithm="giac")

[Out]

1/27*log(abs(sqrt(3)*x + 2*sqrt(b)))/b - 1/27*log(abs(-sqrt(3)*x + sqrt(b)))/b - 1/9/((sqrt(3)*x - sqrt(b))*sq
rt(b))