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]

-2/9*(3-8*x)/(-4*x^2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = 0.00, 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))])

________________________________________________________________________________________

fricas [A]  time = 0.96, size = 29, normalized size = 1.32 \[ -\frac {2 \, \sqrt {-4 \, x^{2} + 3 \, x} {\left (8 \, x - 3\right )}}{9 \, {\left (4 \, x^{2} - 3 \, x\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.51, size = 29, normalized size = 1.32 \[ -\frac {2 \, \sqrt {-4 \, x^{2} + 3 \, x} {\left (8 \, x - 3\right )}}{9 \, {\left (4 \, x^{2} - 3 \, x\right )}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 25, normalized size = 1.14 \[ -\frac {2 \left (4 x -3\right ) \left (8 x -3\right ) x}{9 \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 28, normalized size = 1.27 \[ \frac {16 \, x}{9 \, \sqrt {-4 \, x^{2} + 3 \, x}} - \frac {2}{3 \, \sqrt {-4 \, x^{2} + 3 \, x}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.14, size = 18, normalized size = 0.82 \[ \frac {16\,x-6}{9\,\sqrt {3\,x-4\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x - 4*x^2)^(3/2),x)

[Out]

(16*x - 6)/(9*(3*x - 4*x^2)^(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- 4 x^{2} + 3 x\right )^{\frac {3}{2}}}\, dx \]

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)

________________________________________________________________________________________