3.22 \(\int \frac{1}{(3 x-4 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{2 (3-8 x)}{9 \sqrt{3 x-4 x^2}} \]

[Out]

(-2*(3 - 8*x))/(9*Sqrt[3*x - 4*x^2])

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Rubi [A]  time = 0.0023807, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {613} \[ -\frac{2 (3-8 x)}{9 \sqrt{3 x-4 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(3*x - 4*x^2)^(-3/2),x]

[Out]

(-2*(3 - 8*x))/(9*Sqrt[3*x - 4*x^2])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (3 x-4 x^2\right )^{3/2}} \, dx &=-\frac{2 (3-8 x)}{9 \sqrt{3 x-4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0057725, size = 21, normalized size = 0.95 \[ \frac{2 (8 x-3)}{9 \sqrt{-x (4 x-3)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3*x - 4*x^2)^(-3/2),x]

[Out]

(2*(-3 + 8*x))/(9*Sqrt[-(x*(-3 + 4*x))])

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Maple [A]  time = 0.047, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,x \left ( -3+4\,x \right ) \left ( -3+8\,x \right ) }{9} \left ( -4\,{x}^{2}+3\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*x^2+3*x)^(3/2),x)

[Out]

-2/9*x*(-3+4*x)*(-3+8*x)/(-4*x^2+3*x)^(3/2)

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Maxima [A]  time = 1.16826, size = 38, normalized size = 1.73 \begin{align*} \frac{16 \, x}{9 \, \sqrt{-4 \, x^{2} + 3 \, x}} - \frac{2}{3 \, \sqrt{-4 \, x^{2} + 3 \, x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(3/2),x, algorithm="maxima")

[Out]

16/9*x/sqrt(-4*x^2 + 3*x) - 2/3/sqrt(-4*x^2 + 3*x)

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Fricas [A]  time = 2.16481, size = 66, normalized size = 3. \begin{align*} -\frac{2 \, \sqrt{-4 \, x^{2} + 3 \, x}{\left (8 \, x - 3\right )}}{9 \,{\left (4 \, x^{2} - 3 \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(3/2),x, algorithm="fricas")

[Out]

-2/9*sqrt(-4*x^2 + 3*x)*(8*x - 3)/(4*x^2 - 3*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x**2+3*x)**(3/2),x)

[Out]

Integral((-4*x**2 + 3*x)**(-3/2), x)

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Giac [A]  time = 1.32198, size = 39, normalized size = 1.77 \begin{align*} -\frac{2 \, \sqrt{-4 \, x^{2} + 3 \, x}{\left (8 \, x - 3\right )}}{9 \,{\left (4 \, x^{2} - 3 \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(3/2),x, algorithm="giac")

[Out]

-2/9*sqrt(-4*x^2 + 3*x)*(8*x - 3)/(4*x^2 - 3*x)