3.223 \(\int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac {3 (x-1) (x+1)}{2 \sqrt [3]{(x-1)^4 (x+1)^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6719, 37} \[ \frac {3 (1-x) (x+1)}{2 \sqrt [3]{(1-x)^4 (x+1)^2}} \]

Antiderivative was successfully verified.

[In]

Int[((-1 + x)^4*(1 + x)^2)^(-1/3),x]

[Out]

(3*(1 - x)*(1 + x))/(2*((1 - x)^4*(1 + x)^2)^(1/3))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ -\frac {3 (x-1) (x+1)}{2 \sqrt [3]{(x-1)^4 (x+1)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-1 + x)*(1 + x))/(2*((-1 + x)^4*(1 + x)^2)^(1/3))

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IntegrateAlgebraic [A]  time = 0.42, size = 46, normalized size = 1.84 \[ -\frac {3 \left (x^6-2 x^5-x^4+4 x^3-x^2-2 x+1\right )^{2/3}}{2 (x-1)^3 (x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(1 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6)^(2/3))/(2*(-1 + x)^3*(1 + x))

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fricas [B]  time = 0.94, size = 47, normalized size = 1.88 \[ -\frac {3 \, {\left (x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, x + 1\right )}^{\frac {2}{3}}}{2 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1)^(2/3)/(x^4 - 2*x^3 + 2*x - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left ({\left (x + 1\right )}^{2} {\left (x - 1\right )}^{4}\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((x + 1)^2*(x - 1)^4)^(-1/3), x)

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maple [A]  time = 0.04, size = 22, normalized size = 0.88

method	result	size
gosper	\(-\frac {3 \left (-1+x \right ) \left (1+x \right )}{2 \left (\left (-1+x \right )^{4} \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\)	\(22\)
risch	\(-\frac {3 \left (-1+x \right ) \left (1+x \right )}{2 \left (\left (-1+x \right )^{4} \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\)	\(22\)
trager	\(-\frac {3 \left (x^{6}-2 x^{5}-x^{4}+4 x^{3}-x^{2}-2 x +1\right )^{\frac {2}{3}}}{2 \left (1+x \right ) \left (-1+x \right )^{3}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(-1+x)*(1+x)/((-1+x)^4*(1+x)^2)^(1/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left ({\left (x + 1\right )}^{2} {\left (x - 1\right )}^{4}\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((x + 1)^2*(x - 1)^4)^(-1/3), x)

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mupad [B]  time = 0.24, size = 25, normalized size = 1.00 \[ -\frac {3\,{\left ({\left (x-1\right )}^4\,{\left (x+1\right )}^2\right )}^{2/3}}{2\,{\left (x-1\right )}^3\,\left (x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x - 1)^4*(x + 1)^2)^(1/3),x)

[Out]

-(3*((x - 1)^4*(x + 1)^2)^(2/3))/(2*(x - 1)^3*(x + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{\left (x - 1\right )^{4} \left (x + 1\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)**4*(1+x)**2)**(1/3),x)

[Out]

Integral(((x - 1)**4*(x + 1)**2)**(-1/3), x)

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