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

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

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Rubi [A]  time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {39} \begin {gather*} \frac {x}{18 \sqrt {2} \sqrt {3-b x} \sqrt {b x+3}} \end {gather*}

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])

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}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx &=\frac {x}{18 \sqrt {2} \sqrt {3-b x} \sqrt {3+b x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.66 \begin {gather*} \frac {x}{18 \sqrt {18-2 b^2 x^2}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.21, size = 29, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b x + 3} \sqrt {-2 \, b x + 6} x}{36 \, {\left (b^{2} x^{2} - 9\right )}} \end {gather*}

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, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.14, size = 91, normalized size = 3.14 \begin {gather*} \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 )}} \end {gather*}

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, 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)))

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maple [A]  time = 0.00, size = 24, normalized size = 0.83 \begin {gather*} -\frac {\left (b x -3\right ) x}{9 \sqrt {b x +3}\, \left (-2 b x +6\right )^{\frac {3}{2}}} \end {gather*}

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 = 1.25, size = 15, normalized size = 0.52 \begin {gather*} \frac {x}{18 \, \sqrt {-2 \, b^{2} x^{2} + 18}} \end {gather*}

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, algorithm="maxima")

[Out]

1/18*x/sqrt(-2*b^2*x^2 + 18)

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mupad [B]  time = 0.32, size = 26, normalized size = 0.90 \begin {gather*} -\frac {x\,\sqrt {6-2\,b\,x}}{\sqrt {b\,x+3}\,\left (36\,b\,x-108\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*(6 - 2*b*x)^(1/2))/((b*x + 3)^(1/2)*(36*b*x - 108))

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sympy [C]  time = 31.50, size = 83, normalized size = 2.86 \begin {gather*} - \frac {\sqrt {2} 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 )}}{72 \pi ^{\frac {3}{2}} b} + \frac {\sqrt {2} {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 )}}{72 \pi ^{\frac {3}{2}} b} \end {gather*}

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]

-sqrt(2)*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))/(72*pi**(3/2)
*b) + sqrt(2)*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))/(72*pi**(3/2)*b)

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