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

Optimal. Leaf size=15 \[ \frac{3}{4} \sqrt [3]{x^4-4 x} \]

[Out]

(3*(-4*x + x^4)^(1/3))/4

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Rubi [A]  time = 0.00760728, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{3}{4} \sqrt [3]{x^4-4 x} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^3)/(-4*x + x^4)^(2/3),x]

[Out]

(3*(-4*x + x^4)^(1/3))/4

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Rubi in Sympy [A]  time = 1.29084, size = 12, normalized size = 0.8 \[ \frac{3 \sqrt [3]{x^{4} - 4 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(x**4 - 4*x)**(1/3)/4

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Mathematica [A]  time = 0.01844, size = 15, normalized size = 1. \[ \frac{3}{4} \sqrt [3]{x \left (x^3-4\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^3)/(-4*x + x^4)^(2/3),x]

[Out]

(3*(x*(-4 + x^3))^(1/3))/4

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Maple [A]  time = 0.01, size = 18, normalized size = 1.2 \[{\frac{3\,x \left ({x}^{3}-4 \right ) }{4} \left ({x}^{4}-4\,x \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-1)/(x^4-4*x)^(2/3),x)

[Out]

3/4*x*(x^3-4)/(x^4-4*x)^(2/3)

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Maxima [A]  time = 0.724625, size = 15, normalized size = 1. \[ \frac{3}{4} \,{\left (x^{4} - 4 \, x\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(x^4 - 4*x)^(1/3)

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Fricas [A]  time = 0.259559, size = 15, normalized size = 1. \[ \frac{3}{4} \,{\left (x^{4} - 4 \, x\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(x^4 - 4*x)^(1/3)

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Sympy [A]  time = 0.518973, size = 12, normalized size = 0.8 \[ \frac{3 \sqrt [3]{x^{4} - 4 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(x**4 - 4*x)**(1/3)/4

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GIAC/XCAS [A]  time = 0.284484, size = 15, normalized size = 1. \[ \frac{3}{4} \,{\left (x^{4} - 4 \, x\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*(x^4 - 4*x)^(1/3)