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 = 0.0025479, 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, Rules used = {39} \[ \frac{x}{18 \sqrt{2} \sqrt{3-b x} \sqrt{b x+3}} \]

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])

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*}

Mathematica [A]  time = 0.0177079, 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 = 0.002, size = 24, normalized size = 0.8 \begin{align*} -{\frac{ \left ( bx-3 \right ) x}{9}{\frac{1}{\sqrt{bx+3}}} \left ( -2\,bx+6 \right ) ^{-{\frac{3}{2}}}} \end{align*}

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 = 0.969727, size = 20, normalized size = 0.69 \begin{align*} \frac{x}{18 \, \sqrt{-2 \, b^{2} x^{2} + 18}} \end{align*}

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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Fricas [A]  time = 1.59389, size = 73, normalized size = 2.52 \begin{align*} -\frac{\sqrt{b x + 3} \sqrt{-2 \, b x + 6} x}{36 \,{\left (b^{2} x^{2} - 9\right )}} \end{align*}

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

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 [B]  time = 1.07303, size = 123, normalized size = 4.24 \begin{align*} \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{align*}

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)))