3.7.2 \(\int \frac {-1+x^3}{(-4 x+x^4)^{2/3}} \, dx\) [602]

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

[Out]

3/4*(x^4-4*x)^(1/3)

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {1602} \begin {gather*} \frac {3}{4} \sqrt [3]{x^4-4 x} \end {gather*}

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 1602

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*}

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Mathematica [A]
time = 10.04, size = 15, normalized size = 1.00 \begin {gather*} \frac {3}{4} \sqrt [3]{x \left (-4+x^3\right )} \end {gather*}

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 = 0.32, size = 12, normalized size = 0.80

method	result	size
default	\(\frac {3 \left (x^{4}-4 x \right )^{\frac {1}{3}}}{4}\)	\(12\)
trager	\(\frac {3 \left (x^{4}-4 x \right )^{\frac {1}{3}}}{4}\)	\(12\)
gosper	\(\frac {3 x \left (x^{3}-4\right )}{4 \left (x^{4}-4 x \right )^{\frac {2}{3}}}\)	\(18\)
risch	\(\frac {3 x \left (x^{3}-4\right )}{4 \left (x \left (x^{3}-4\right )\right )^{\frac {2}{3}}}\)	\(18\)
meijerg	\(-\frac {3 \,2^{\frac {2}{3}} \left (-\mathrm {signum}\left (-1+\frac {x^{3}}{4}\right )\right )^{\frac {2}{3}} x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{9}, \frac {2}{3}\right ], \left [\frac {10}{9}\right ], \frac {x^{3}}{4}\right )}{4 \mathrm {signum}\left (-1+\frac {x^{3}}{4}\right )^{\frac {2}{3}}}+\frac {3 \,2^{\frac {2}{3}} \left (-\mathrm {signum}\left (-1+\frac {x^{3}}{4}\right )\right )^{\frac {2}{3}} x^{\frac {10}{3}} \hypergeom \left (\left [\frac {2}{3}, \frac {10}{9}\right ], \left [\frac {19}{9}\right ], \frac {x^{3}}{4}\right )}{40 \mathrm {signum}\left (-1+\frac {x^{3}}{4}\right )^{\frac {2}{3}}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)/(x^4-4*x)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3/4*(x^4-4*x)^(1/3)

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Maxima [A]
time = 0.27, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{4} \, {\left (x^{4} - 4 \, x\right )}^{\frac {1}{3}} \end {gather*}

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 = 0.33, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{4} \, {\left (x^{4} - 4 \, x\right )}^{\frac {1}{3}} \end {gather*}

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 = 0.07, size = 12, normalized size = 0.80 \begin {gather*} \frac {3 \sqrt [3]{x^{4} - 4 x}}{4} \end {gather*}

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 = 2.81, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{4} \, {\left (x^{4} - 4 \, x\right )}^{\frac {1}{3}} \end {gather*}

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)

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Mupad [B]
time = 3.52, size = 11, normalized size = 0.73 \begin {gather*} \frac {3\,{\left (x^4-4\,x\right )}^{1/3}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - 1)/(x^4 - 4*x)^(2/3),x)

[Out]

(3*(x^4 - 4*x)^(1/3))/4

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