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

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 = 0.0202536, 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 \[ -\frac{64 (3-8 x)}{243 \sqrt{3 x-4 x^2}}-\frac{2 (3-8 x)}{27 \left (3 x-4 x^2\right )^{3/2}} \]

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])

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Rubi in Sympy [A]  time = 1.61603, size = 37, normalized size = 0.82 \[ - \frac{32 \left (- 16 x + 6\right )}{243 \sqrt{- 4 x^{2} + 3 x}} - \frac{2 \left (- 8 x + 3\right )}{27 \left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-4*x**2+3*x)**(5/2),x)

[Out]

-32*(-16*x + 6)/(243*sqrt(-4*x**2 + 3*x)) - 2*(-8*x + 3)/(27*(-4*x**2 + 3*x)**(3
/2))

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Mathematica [A]  time = 0.0230932, 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 = 0.004, size = 35, normalized size = 0.8 \[{\frac{2\,x \left ( 4\,x-3 \right ) \left ( 1024\,{x}^{3}-1152\,{x}^{2}+216\,x+27 \right ) }{243} \left ( -4\,{x}^{2}+3\,x \right ) ^{-{\frac{5}{2}}}} \]

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

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Maxima [A]  time = 0.689655, size = 74, normalized size = 1.64 \[ \frac{512 \, x}{243 \, \sqrt{-4 \, x^{2} + 3 \, x}} - \frac{64}{81 \, \sqrt{-4 \, x^{2} + 3 \, x}} + \frac{16 \, x}{27 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}}} - \frac{2}{9 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-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 = 0.216312, size = 53, normalized size = 1.18 \[ \frac{2 \,{\left (1024 \, x^{3} - 1152 \, x^{2} + 216 \, x + 27\right )}}{243 \,{\left (4 \, x^{2} - 3 \, x\right )} \sqrt{-4 \, x^{2} + 3 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- 4 x^{2} + 3 x\right )^{\frac{5}{2}}}\, dx \]

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/XCAS [A]  time = 0.219204, size = 53, normalized size = 1.18 \[ -\frac{2 \,{\left (8 \,{\left (16 \,{\left (8 \, x - 9\right )} x + 27\right )} x + 27\right )} \sqrt{-4 \, x^{2} + 3 \, x}}{243 \,{\left (4 \, x^{2} - 3 \, x\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-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