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

Optimal. Leaf size=26 \[ \frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {x+3}} \]

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {39} \begin {gather*} \frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.81 \begin {gather*} \frac {x}{18 \sqrt {6-2 x} \sqrt {x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(18*Sqrt[6 - 2*x]*Sqrt[3 + x])

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IntegrateAlgebraic [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.29, size = 22, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {x + 3} x \sqrt {-2 \, x + 6}}{36 \, {\left (x^{2} - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.04, size = 71, normalized size = 2.73 \begin {gather*} \frac {\sqrt {2} {\left (\sqrt {6} - \sqrt {-x + 3}\right )}}{144 \, \sqrt {x + 3}} - \frac {\sqrt {2} \sqrt {x + 3} \sqrt {-x + 3}}{72 \, {\left (x - 3\right )}} - \frac {\sqrt {2} \sqrt {x + 3}}{144 \, {\left (\sqrt {6} - \sqrt {-x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 19, normalized size = 0.73 \begin {gather*} -\frac {\left (x -3\right ) x}{9 \sqrt {x +3}\, \left (-2 x +6\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 12, normalized size = 0.46 \begin {gather*} \frac {x}{18 \, \sqrt {-2 \, x^{2} + 18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x/sqrt(-2*x^2 + 18)

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mupad [B]  time = 0.37, size = 22, normalized size = 0.85 \begin {gather*} -\frac {x\,\sqrt {6-2\,x}}{\left (36\,x-108\right )\,\sqrt {x+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 20.45, size = 90, normalized size = 3.46 \begin {gather*} \begin {cases} \frac {\sqrt {2}}{36 \sqrt {-1 + \frac {6}{x + 3}}} - \frac {\sqrt {2}}{12 \sqrt {-1 + \frac {6}{x + 3}} \left (x + 3\right )} & \text {for}\: \frac {6}{\left |{x + 3}\right |} > 1 \\\frac {\sqrt {2} i \sqrt {1 - \frac {6}{x + 3}} \left (x + 3\right )}{108 - 36 x} - \frac {3 \sqrt {2} i \sqrt {1 - \frac {6}{x + 3}}}{108 - 36 x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(6-2*x)**(3/2)/(3+x)**(3/2),x)

[Out]

Piecewise((sqrt(2)/(36*sqrt(-1 + 6/(x + 3))) - sqrt(2)/(12*sqrt(-1 + 6/(x + 3))*(x + 3)), 6/Abs(x + 3) > 1), (
sqrt(2)*I*sqrt(1 - 6/(x + 3))*(x + 3)/(108 - 36*x) - 3*sqrt(2)*I*sqrt(1 - 6/(x + 3))/(108 - 36*x), True))

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