3.1.65 \(\int \frac {-1+3 x^2}{\sqrt [3]{-x+x^3}} \, dx\)

Optimal. Leaf size=15 \[ \frac {3}{2} \left (x^3-x\right )^{2/3} \]

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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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1588} \begin {gather*} \frac {3}{2} \left (x^3-x\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x^2)/(-x + x^3)^(1/3),x]

[Out]

(3*(-x + x^3)^(2/3))/2

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

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Mathematica [A]  time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {3}{2} \left (x \left (x^2-1\right )\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x*(-1 + x^2))^(2/3))/2

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IntegrateAlgebraic [A]  time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {3}{2} \left (-x+x^3\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 3*x^2)/(-x + x^3)^(1/3),x]

[Out]

(3*(-x + x^3)^(2/3))/2

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fricas [A]  time = 0.43, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{2} \, {\left (x^{3} - x\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^3 - x)^(2/3)

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giac [A]  time = 0.29, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{2} \, {\left (x^{3} - x\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^3 - x)^(2/3)

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maple [A]  time = 0.08, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}}}{2}\)	\(12\)
default	\(\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}}}{2}\)	\(12\)
trager	\(\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}}}{2}\)	\(12\)
risch	\(\frac {3 x \left (x^{2}-1\right )}{2 \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}\)	\(18\)
gosper	\(\frac {3 \left (1+x \right ) \left (-1+x \right ) x}{2 \left (x^{3}-x \right )^{\frac {1}{3}}}\)	\(19\)
meijerg	\(-\frac {3 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right ) x^{\frac {2}{3}}}{2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}+\frac {9 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x^{2}\right ) x^{\frac {8}{3}}}{8 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{2} \, {\left (x^{3} - x\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^3 - x)^(2/3)

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mupad [B]  time = 0.29, size = 11, normalized size = 0.73 \begin {gather*} \frac {3\,{\left (x^3-x\right )}^{2/3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 - 1)/(x^3 - x)^(1/3),x)

[Out]

(3*(x^3 - x)^(2/3))/2

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sympy [A]  time = 0.14, size = 10, normalized size = 0.67 \begin {gather*} \frac {3 \left (x^{3} - x\right )^{\frac {2}{3}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2-1)/(x**3-x)**(1/3),x)

[Out]

3*(x**3 - x)**(2/3)/2

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