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 = 0.0200175, 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 \[ \frac{x}{9 \sqrt{3-b x} \sqrt{b x+3}} \]

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])

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Rubi in Sympy [A]  time = 3.84959, size = 19, normalized size = 0.79 \[ \frac{x}{9 \sqrt{- b x + 3} \sqrt{b x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(9*sqrt(-b*x + 3)*sqrt(b*x + 3))

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Mathematica [A]  time = 0.0190428, 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 = 0.004, size = 19, normalized size = 0.8 \[{\frac{x}{9}{\frac{1}{\sqrt{-bx+3}}}{\frac{1}{\sqrt{bx+3}}}} \]

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 = 1.35052, size = 20, normalized size = 0.83 \[ \frac{x}{9 \, \sqrt{-b^{2} x^{2} + 9}} \]

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 = 0.201369, size = 69, normalized size = 2.88 \[ -\frac{\sqrt{b x + 3} \sqrt{-b x + 3} x - 3 \, x}{9 \,{\left (b^{2} x^{2} + 3 \, \sqrt{b x + 3} \sqrt{-b x + 3} - 9\right )}} \]

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

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Sympy [A]  time = 20.4051, size = 73, normalized size = 3.04 \[ - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{9}{b^{2} x^{2}}} \right )}}{18 \pi ^{\frac{3}{2}} b} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{9 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{18 \pi ^{\frac{3}{2}} b} \]

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) + meijerg(((-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/XCAS [A]  time = 0.21655, size = 111, normalized size = 4.62 \[ \frac{\sqrt{6} - \sqrt{-b x + 3}}{36 \, \sqrt{b x + 3} b} - \frac{\sqrt{b x + 3} \sqrt{-b x + 3}}{18 \,{\left (b x - 3\right )} b} - \frac{\sqrt{b x + 3}}{36 \, b{\left (\sqrt{6} - \sqrt{-b x + 3}\right )}} \]

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*sqrt(b*x + 3)/(b*(sqrt(6) - sqrt(-b*x + 3)))