3.1166 \(\int \frac{1}{(3-b x)^{3/2} (3+b x)^{3/2}} \, dx\)

Optimal. Leaf size=24 \[ \frac{x}{9 \sqrt{3-b x} \sqrt{b x+3}} \]

[Out]

x/(9*Sqrt[3 - b*x]*Sqrt[3 + b*x])

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Rubi [A]  time = 0.0025522, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {39} \[ \frac{x}{9 \sqrt{3-b x} \sqrt{b x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(9*Sqrt[3 - b*x]*Sqrt[3 + b*x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(3-b x)^{3/2} (3+b x)^{3/2}} \, dx &=\frac{x}{9 \sqrt{3-b x} \sqrt{3+b x}}\\ \end{align*}

Mathematica [A]  time = 0.007636, size = 19, normalized size = 0.79 \[ \frac{x}{9 \sqrt{9-b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(9*Sqrt[9 - b^2*x^2])

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Maple [A]  time = 0.003, size = 19, normalized size = 0.8 \begin{align*}{\frac{x}{9}{\frac{1}{\sqrt{-bx+3}}}{\frac{1}{\sqrt{bx+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.995133, size = 20, normalized size = 0.83 \begin{align*} \frac{x}{9 \, \sqrt{-b^{2} x^{2} + 9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x/sqrt(-b^2*x^2 + 9)

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Fricas [A]  time = 1.55026, size = 69, normalized size = 2.88 \begin{align*} -\frac{\sqrt{b x + 3} \sqrt{-b x + 3} x}{9 \,{\left (b^{2} x^{2} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 5.32864, size = 73, normalized size = 3.04 \begin{align*} - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{9}{b^{2} x^{2}}} \right )}}{18 \pi ^{\frac{3}{2}} b} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{9 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{18 \pi ^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*meijerg(((3/4, 5/4, 1), (1/2, 3/2, 2)), ((3/4, 1, 5/4, 3/2, 2), (0,)), 9/(b**2*x**2))/(18*pi**(3/2)*b) + me
ijerg(((-1/2, 0, 1/4, 1/2, 3/4, 1), ()), ((1/4, 3/4), (-1/2, 0, 1, 0)), 9*exp_polar(-2*I*pi)/(b**2*x**2))/(18*
pi**(3/2)*b)

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Giac [B]  time = 1.06276, size = 111, normalized size = 4.62 \begin{align*} \frac{\sqrt{6} - \sqrt{-b x + 3}}{36 \, \sqrt{b x + 3} b} - \frac{\sqrt{b x + 3} \sqrt{-b x + 3}}{18 \,{\left (b x - 3\right )} b} - \frac{\sqrt{b x + 3}}{36 \, b{\left (\sqrt{6} - \sqrt{-b x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(sqrt(6) - sqrt(-b*x + 3))/(sqrt(b*x + 3)*b) - 1/18*sqrt(b*x + 3)*sqrt(-b*x + 3)/((b*x - 3)*b) - 1/36*sqr
t(b*x + 3)/(b*(sqrt(6) - sqrt(-b*x + 3)))