3.1168 \(\int \frac{1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ \frac{x}{18 \sqrt{2} \sqrt{3-b x} \sqrt{b x+3}} \]

[Out]

x/(18*Sqrt[2]*Sqrt[3 - b*x]*Sqrt[3 + b*x])

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Rubi [A]  time = 0.0229281, antiderivative size = 29, 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 \[ \frac{x}{18 \sqrt{2} \sqrt{3-b x} \sqrt{b x+3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((6 - 2*b*x)^(3/2)*(3 + b*x)^(3/2)),x]

[Out]

x/(18*Sqrt[2]*Sqrt[3 - b*x]*Sqrt[3 + b*x])

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Rubi in Sympy [A]  time = 3.72135, size = 20, normalized size = 0.69 \[ \frac{x}{18 \sqrt{- 2 b x + 6} \sqrt{b x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-2*b*x+6)**(3/2)/(b*x+3)**(3/2),x)

[Out]

x/(18*sqrt(-2*b*x + 6)*sqrt(b*x + 3))

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Mathematica [A]  time = 0.0547824, size = 19, normalized size = 0.66 \[ \frac{x}{18 \sqrt{18-2 b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((6 - 2*b*x)^(3/2)*(3 + b*x)^(3/2)),x]

[Out]

x/(18*Sqrt[18 - 2*b^2*x^2])

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Maple [A]  time = 0.005, size = 24, normalized size = 0.8 \[ -{\frac{ \left ( bx-3 \right ) x}{9}{\frac{1}{\sqrt{bx+3}}} \left ( -2\,bx+6 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*b*x+6)^(3/2)/(b*x+3)^(3/2),x)

[Out]

-1/9*(b*x-3)/(b*x+3)^(1/2)*x/(-2*b*x+6)^(3/2)

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Maxima [A]  time = 1.34414, size = 20, normalized size = 0.69 \[ \frac{x}{18 \, \sqrt{-2 \, b^{2} x^{2} + 18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 3)^(3/2)*(-2*b*x + 6)^(3/2)),x, algorithm="maxima")

[Out]

1/18*x/sqrt(-2*b^2*x^2 + 18)

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Fricas [A]  time = 0.202442, size = 39, normalized size = 1.34 \[ -\frac{\sqrt{b x + 3} \sqrt{-2 \, b x + 6} x}{36 \,{\left (b^{2} x^{2} - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 3)^(3/2)*(-2*b*x + 6)^(3/2)),x, algorithm="fricas")

[Out]

-1/36*sqrt(b*x + 3)*sqrt(-2*b*x + 6)*x/(b^2*x^2 - 9)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-2*b*x+6)**(3/2)/(b*x+3)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220371, size = 123, normalized size = 4.24 \[ \frac{\sqrt{2}{\left (\sqrt{6} - \sqrt{-b x + 3}\right )}}{144 \, \sqrt{b x + 3} b} - \frac{\sqrt{2} \sqrt{b x + 3} \sqrt{-b x + 3}}{72 \,{\left (b x - 3\right )} b} - \frac{\sqrt{2} \sqrt{b x + 3}}{144 \, b{\left (\sqrt{6} - \sqrt{-b x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + 3)^(3/2)*(-2*b*x + 6)^(3/2)),x, algorithm="giac")

[Out]

1/144*sqrt(2)*(sqrt(6) - sqrt(-b*x + 3))/(sqrt(b*x + 3)*b) - 1/72*sqrt(2)*sqrt(b
*x + 3)*sqrt(-b*x + 3)/((b*x - 3)*b) - 1/144*sqrt(2)*sqrt(b*x + 3)/(b*(sqrt(6) -
sqrt(-b*x + 3)))