3.24 \(\int \frac{1}{\left (3 x-4 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{4096 (3-8 x)}{10935 \sqrt{3 x-4 x^2}}-\frac{128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac{2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}} \]

[Out]

(-2*(3 - 8*x))/(45*(3*x - 4*x^2)^(5/2)) - (128*(3 - 8*x))/(1215*(3*x - 4*x^2)^(3
/2)) - (4096*(3 - 8*x))/(10935*Sqrt[3*x - 4*x^2])

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Rubi [A]  time = 0.0330418, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{4096 (3-8 x)}{10935 \sqrt{3 x-4 x^2}}-\frac{128 (3-8 x)}{1215 \left (3 x-4 x^2\right )^{3/2}}-\frac{2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(3*x - 4*x^2)^(-7/2),x]

[Out]

(-2*(3 - 8*x))/(45*(3*x - 4*x^2)^(5/2)) - (128*(3 - 8*x))/(1215*(3*x - 4*x^2)^(3
/2)) - (4096*(3 - 8*x))/(10935*Sqrt[3*x - 4*x^2])

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Rubi in Sympy [A]  time = 2.15337, size = 56, normalized size = 0.84 \[ - \frac{2048 \left (- 16 x + 6\right )}{10935 \sqrt{- 4 x^{2} + 3 x}} - \frac{128 \left (- 8 x + 3\right )}{1215 \left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}} - \frac{2 \left (- 8 x + 3\right )}{45 \left (- 4 x^{2} + 3 x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-4*x**2+3*x)**(7/2),x)

[Out]

-2048*(-16*x + 6)/(10935*sqrt(-4*x**2 + 3*x)) - 128*(-8*x + 3)/(1215*(-4*x**2 +
3*x)**(3/2)) - 2*(-8*x + 3)/(45*(-4*x**2 + 3*x)**(5/2))

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Mathematica [A]  time = 0.0317378, size = 51, normalized size = 0.76 \[ \frac{2 \left (262144 x^5-491520 x^4+276480 x^3-34560 x^2-3240 x-729\right )}{10935 (3-4 x)^2 x^2 \sqrt{-x (4 x-3)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3*x - 4*x^2)^(-7/2),x]

[Out]

(2*(-729 - 3240*x - 34560*x^2 + 276480*x^3 - 491520*x^4 + 262144*x^5))/(10935*(3
- 4*x)^2*x^2*Sqrt[-(x*(-3 + 4*x))])

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Maple [A]  time = 0.005, size = 45, normalized size = 0.7 \[ -{\frac{2\,x \left ( 4\,x-3 \right ) \left ( 262144\,{x}^{5}-491520\,{x}^{4}+276480\,{x}^{3}-34560\,{x}^{2}-3240\,x-729 \right ) }{10935} \left ( -4\,{x}^{2}+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-4*x^2+3*x)^(7/2),x)

[Out]

-2/10935*x*(4*x-3)*(262144*x^5-491520*x^4+276480*x^3-34560*x^2-3240*x-729)/(-4*x
^2+3*x)^(7/2)

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Maxima [A]  time = 0.703799, size = 111, normalized size = 1.66 \[ \frac{32768 \, x}{10935 \, \sqrt{-4 \, x^{2} + 3 \, x}} - \frac{4096}{3645 \, \sqrt{-4 \, x^{2} + 3 \, x}} + \frac{1024 \, x}{1215 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}}} - \frac{128}{405 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}}} + \frac{16 \, x}{45 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{5}{2}}} - \frac{2}{15 \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x^2 + 3*x)^(-7/2),x, algorithm="maxima")

[Out]

32768/10935*x/sqrt(-4*x^2 + 3*x) - 4096/3645/sqrt(-4*x^2 + 3*x) + 1024/1215*x/(-
4*x^2 + 3*x)^(3/2) - 128/405/(-4*x^2 + 3*x)^(3/2) + 16/45*x/(-4*x^2 + 3*x)^(5/2)
- 2/15/(-4*x^2 + 3*x)^(5/2)

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Fricas [A]  time = 0.225794, size = 76, normalized size = 1.13 \[ \frac{2 \,{\left (262144 \, x^{5} - 491520 \, x^{4} + 276480 \, x^{3} - 34560 \, x^{2} - 3240 \, x - 729\right )}}{10935 \,{\left (16 \, x^{4} - 24 \, x^{3} + 9 \, x^{2}\right )} \sqrt{-4 \, x^{2} + 3 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x^2 + 3*x)^(-7/2),x, algorithm="fricas")

[Out]

2/10935*(262144*x^5 - 491520*x^4 + 276480*x^3 - 34560*x^2 - 3240*x - 729)/((16*x
^4 - 24*x^3 + 9*x^2)*sqrt(-4*x^2 + 3*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- 4 x^{2} + 3 x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-4*x**2+3*x)**(7/2),x)

[Out]

Integral((-4*x**2 + 3*x)**(-7/2), x)

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GIAC/XCAS [A]  time = 0.220974, size = 66, normalized size = 0.99 \[ -\frac{2 \,{\left (8 \,{\left (32 \,{\left (8 \,{\left (16 \,{\left (8 \, x - 15\right )} x + 135\right )} x - 135\right )} x - 405\right )} x - 729\right )} \sqrt{-4 \, x^{2} + 3 \, x}}{10935 \,{\left (4 \, x^{2} - 3 \, x\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x^2 + 3*x)^(-7/2),x, algorithm="giac")

[Out]

-2/10935*(8*(32*(8*(16*(8*x - 15)*x + 135)*x - 135)*x - 405)*x - 729)*sqrt(-4*x^
2 + 3*x)/(4*x^2 - 3*x)^3