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

Optimal result

Integrand size = 24, antiderivative size = 77 \[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\frac {1}{3 \sqrt {3} \sqrt {b} \left (\sqrt {3} \sqrt {b}-3 x\right )}-\frac {\log \left (\sqrt {b}-\sqrt {3} x\right )}{27 b}+\frac {\log \left (2 \sqrt {b}+\sqrt {3} x\right )}{27 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2088, 46} \[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\frac {1}{3 \sqrt {3} \sqrt {b} \left (\sqrt {3} \sqrt {b}-3 x\right )}-\frac {\log \left (\sqrt {b}-\sqrt {3} x\right )}{27 b}+\frac {\log \left (2 \sqrt {b}+\sqrt {3} x\right )}{27 b} \]

[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 46

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] && Lt
Q[m + n + 2, 0])

Rule 2088

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.86 \[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=-\frac {\left (-\sqrt {3} \sqrt {b}+3 x\right ) \left (2 \sqrt {3} \sqrt {b}+3 x\right ) \left (3 \sqrt {3} \sqrt {b}+\left (-\sqrt {3} \sqrt {b}+3 x\right ) \log \left (-\sqrt {3} \sqrt {b}+3 x\right )+\left (\sqrt {3} \sqrt {b}-3 x\right ) \log \left (2 \sqrt {3} \sqrt {b}+3 x\right )\right )}{81 b \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )} \]

[In]

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

[Out]

-1/81*((-(Sqrt[3]*Sqrt[b]) + 3*x)*(2*Sqrt[3]*Sqrt[b] + 3*x)*(3*Sqrt[3]*Sqrt[b] + (-(Sqrt[3]*Sqrt[b]) + 3*x)*Lo
g[-(Sqrt[3]*Sqrt[b]) + 3*x] + (Sqrt[3]*Sqrt[b] - 3*x)*Log[2*Sqrt[3]*Sqrt[b] + 3*x]))/(b*(2*Sqrt[3]*b^(3/2) - 9
*b*x + 9*x^3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-9 b \textit {\_Z} +9 \textit {\_Z}^{3}+2 b^{\frac {3}{2}} \sqrt {3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-b}\right )}{9}\)	\(43\)

[In]

int(1/(-9*b*x+9*x^3+2*b^(3/2)*3^(1/2)),x,method=_RETURNVERBOSE)

[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)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=-\frac {3 \, \sqrt {3} \sqrt {b} x - {\left (3 \, x^{2} - b\right )} \log \left (2 \, \sqrt {3} \sqrt {b} + 3 \, x\right ) + {\left (3 \, x^{2} - b\right )} \log \left (-\sqrt {3} \sqrt {b} + 3 \, x\right ) + 3 \, b}{27 \, {\left (3 \, b x^{2} - b^{2}\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=- \frac {3 \sqrt {3}}{81 \sqrt {b} x - 27 \sqrt {3} b} + \frac {- \frac {\log {\left (- \frac {\sqrt {3} \sqrt {b}}{3} + x \right )}}{27} + \frac {\log {\left (\frac {2 \sqrt {3} \sqrt {b}}{3} + x \right )}}{27}}{b} \]

[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

Maxima [F]

\[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\int { \frac {1}{9 \, x^{3} + 2 \, \sqrt {3} b^{\frac {3}{2}} - 9 \, b x} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\frac {\log \left ({\left | 9 \, \sqrt {3} x + 18 \, \sqrt {b} \right |}\right )}{27 \, b} - \frac {\log \left ({\left | -\sqrt {3} x + \sqrt {b} \right |}\right )}{27 \, b} - \frac {1}{9 \, {\left (\sqrt {3} x - \sqrt {b}\right )} \sqrt {b}} \]

[In]

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

[Out]

1/27*log(abs(9*sqrt(3)*x + 18*sqrt(b)))/b - 1/27*log(abs(-sqrt(3)*x + sqrt(b)))/b - 1/9/((sqrt(3)*x - sqrt(b))
*sqrt(b))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66 \[ \int \frac {1}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\frac {2\,\sqrt {3}\,\sqrt {27}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sqrt {27}}{27}+\frac {2\,\sqrt {27}\,x}{9\,\sqrt {b}}\right )}{243\,b}-\frac {\sqrt {3}}{27\,\sqrt {b}\,\left (x-\frac {\sqrt {3}\,\sqrt {b}}{3}\right )} \]

[In]

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

[Out]

(2*3^(1/2)*27^(1/2)*atanh((3^(1/2)*27^(1/2))/27 + (2*27^(1/2)*x)/(9*b^(1/2))))/(243*b) - 3^(1/2)/(27*b^(1/2)*(
x - (3^(1/2)*b^(1/2))/3))