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

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 = 0.0093661, 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, Rules used = {1588} \[ \frac{3}{4} \sqrt [3]{x^4-4 x} \]

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

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{-1+x^3}{\left (-4 x+x^4\right )^{2/3}} \, dx &=\frac{3}{4} \sqrt [3]{-4 x+x^4}\\ \end{align*}

Mathematica [A]  time = 0.0256064, 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 = 0.006, size = 18, normalized size = 1.2 \begin{align*}{\frac{3\,x \left ({x}^{3}-4 \right ) }{4} \left ({x}^{4}-4\,x \right ) ^{-{\frac{2}{3}}}} \end{align*}

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 = 1.10455, size = 15, normalized size = 1. \begin{align*} \frac{3}{4} \,{\left (x^{4} - 4 \, x\right )}^{\frac{1}{3}} \end{align*}

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 = 1.36459, size = 31, normalized size = 2.07 \begin{align*} \frac{3}{4} \,{\left (x^{4} - 4 \, x\right )}^{\frac{1}{3}} \end{align*}

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 = 0.185364, size = 12, normalized size = 0.8 \begin{align*} \frac{3 \sqrt [3]{x^{4} - 4 x}}{4} \end{align*}

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 [A]  time = 1.10566, size = 15, normalized size = 1. \begin{align*} \frac{3}{4} \,{\left (x^{4} - 4 \, x\right )}^{\frac{1}{3}} \end{align*}

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)