3.23 \(\int \frac{1}{(3 x-4 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{64 (3-8 x)}{243 \sqrt{3 x-4 x^2}}-\frac{2 (3-8 x)}{27 \left (3 x-4 x^2\right )^{3/2}} \]

[Out]

(-2*(3 - 8*x))/(27*(3*x - 4*x^2)^(3/2)) - (64*(3 - 8*x))/(243*Sqrt[3*x - 4*x^2])

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Rubi [A]  time = 0.0062287, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {614, 613} \[ -\frac{64 (3-8 x)}{243 \sqrt{3 x-4 x^2}}-\frac{2 (3-8 x)}{27 \left (3 x-4 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(3*x - 4*x^2)^(-5/2),x]

[Out]

(-2*(3 - 8*x))/(27*(3*x - 4*x^2)^(3/2)) - (64*(3 - 8*x))/(243*Sqrt[3*x - 4*x^2])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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 )^{5/2}} \, dx &=-\frac{2 (3-8 x)}{27 \left (3 x-4 x^2\right )^{3/2}}+\frac{32}{27} \int \frac{1}{\left (3 x-4 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (3-8 x)}{27 \left (3 x-4 x^2\right )^{3/2}}-\frac{64 (3-8 x)}{243 \sqrt{3 x-4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0101693, size = 31, normalized size = 0.69 \[ -\frac{2048 x^3-2304 x^2+432 x+54}{243 (-x (4 x-3))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3*x - 4*x^2)^(-5/2),x]

[Out]

-(54 + 432*x - 2304*x^2 + 2048*x^3)/(243*(-(x*(-3 + 4*x)))^(3/2))

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Maple [A]  time = 0.048, size = 35, normalized size = 0.8 \begin{align*}{\frac{2\,x \left ( -3+4\,x \right ) \left ( 1024\,{x}^{3}-1152\,{x}^{2}+216\,x+27 \right ) }{243} \left ( -4\,{x}^{2}+3\,x \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*x^2+3*x)^(5/2),x)

[Out]

2/243*x*(-3+4*x)*(1024*x^3-1152*x^2+216*x+27)/(-4*x^2+3*x)^(5/2)

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Maxima [A]  time = 1.2355, size = 74, normalized size = 1.64 \begin{align*} \frac{512 \, x}{243 \, \sqrt{-4 \, x^{2} + 3 \, x}} - \frac{64}{81 \, \sqrt{-4 \, x^{2} + 3 \, x}} + \frac{16 \, x}{27 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}}} - \frac{2}{9 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(5/2),x, algorithm="maxima")

[Out]

512/243*x/sqrt(-4*x^2 + 3*x) - 64/81/sqrt(-4*x^2 + 3*x) + 16/27*x/(-4*x^2 + 3*x)^(3/2) - 2/9/(-4*x^2 + 3*x)^(3
/2)

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Fricas [A]  time = 2.19676, size = 119, normalized size = 2.64 \begin{align*} -\frac{2 \,{\left (1024 \, x^{3} - 1152 \, x^{2} + 216 \, x + 27\right )} \sqrt{-4 \, x^{2} + 3 \, x}}{243 \,{\left (16 \, x^{4} - 24 \, x^{3} + 9 \, x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(5/2),x, algorithm="fricas")

[Out]

-2/243*(1024*x^3 - 1152*x^2 + 216*x + 27)*sqrt(-4*x^2 + 3*x)/(16*x^4 - 24*x^3 + 9*x^2)

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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{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x**2+3*x)**(5/2),x)

[Out]

Integral((-4*x**2 + 3*x)**(-5/2), x)

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Giac [A]  time = 1.35491, size = 53, normalized size = 1.18 \begin{align*} -\frac{2 \,{\left (8 \,{\left (16 \,{\left (8 \, x - 9\right )} x + 27\right )} x + 27\right )} \sqrt{-4 \, x^{2} + 3 \, x}}{243 \,{\left (4 \, x^{2} - 3 \, x\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(5/2),x, algorithm="giac")

[Out]

-2/243*(8*(16*(8*x - 9)*x + 27)*x + 27)*sqrt(-4*x^2 + 3*x)/(4*x^2 - 3*x)^2