3.225 \(\int \frac {1}{\sqrt [3]{(-1+x)^7 (1+x)^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {9 (x-1)^2 (x+1)}{16 \sqrt [3]{(x-1)^7 (x+1)^2}}-\frac {3 (x-1) (x+1)}{8 \sqrt [3]{(x-1)^7 (x+1)^2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6719, 45, 37} \[ \frac {9 (x+1) (1-x)^2}{16 \sqrt [3]{-(1-x)^7 (x+1)^2}}+\frac {3 (x+1) (1-x)}{8 \sqrt [3]{-(1-x)^7 (x+1)^2}} \]

Antiderivative was successfully verified.

[In]

Int[((-1 + x)^7*(1 + x)^2)^(-1/3),x]

[Out]

(3*(1 - x)*(1 + x))/(8*(-((1 - x)^7*(1 + x)^2))^(1/3)) + (9*(1 - x)^2*(1 + x))/(16*(-((1 - x)^7*(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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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)^7 (1+x)^2}} \, dx &=\frac {\left ((-1+x)^{7/3} (1+x)^{2/3}\right ) \int \frac {1}{(-1+x)^{7/3} (1+x)^{2/3}} \, dx}{\sqrt [3]{(-1+x)^7 (1+x)^2}}\\ &=\frac {3 (1-x) (1+x)}{8 \sqrt [3]{-(1-x)^7 (1+x)^2}}-\frac {\left (3 (-1+x)^{7/3} (1+x)^{2/3}\right ) \int \frac {1}{(-1+x)^{4/3} (1+x)^{2/3}} \, dx}{8 \sqrt [3]{(-1+x)^7 (1+x)^2}}\\ &=\frac {3 (1-x) (1+x)}{8 \sqrt [3]{-(1-x)^7 (1+x)^2}}+\frac {9 (1-x)^2 (1+x)}{16 \sqrt [3]{-(1-x)^7 (1+x)^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.57 \[ \frac {3 (x-1) (x+1) (3 x-5)}{16 \sqrt [3]{(x-1)^7 (x+1)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x)^7*(1 + x)^2)^(-1/3),x]

[Out]

(3*(-1 + x)*(1 + x)*(-5 + 3*x))/(16*((-1 + x)^7*(1 + x)^2)^(1/3))

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IntegrateAlgebraic [A]  time = 2.52, size = 56, normalized size = 1.06 \[ \frac {3 (3 x-5) \left (x^9-5 x^8+8 x^7-14 x^5+14 x^4-8 x^2+5 x-1\right )^{2/3}}{16 (x-1)^6 (x+1)} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x)^7*(1 + x)^2)^(-1/3),x]

[Out]

(3*(-5 + 3*x)*(-1 + 5*x - 8*x^2 + 14*x^4 - 14*x^5 + 8*x^7 - 5*x^8 + x^9)^(2/3))/(16*(-1 + x)^6*(1 + x))

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fricas [A]  time = 0.91, size = 77, normalized size = 1.45 \[ \frac {3 \, {\left (x^{9} - 5 \, x^{8} + 8 \, x^{7} - 14 \, x^{5} + 14 \, x^{4} - 8 \, x^{2} + 5 \, x - 1\right )}^{\frac {2}{3}} {\left (3 \, x - 5\right )}}{16 \, {\left (x^{7} - 5 \, x^{6} + 9 \, x^{5} - 5 \, x^{4} - 5 \, x^{3} + 9 \, x^{2} - 5 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)^7*(1+x)^2)^(1/3),x, algorithm="fricas")

[Out]

3/16*(x^9 - 5*x^8 + 8*x^7 - 14*x^5 + 14*x^4 - 8*x^2 + 5*x - 1)^(2/3)*(3*x - 5)/(x^7 - 5*x^6 + 9*x^5 - 5*x^4 -
5*x^3 + 9*x^2 - 5*x + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left ({\left (x + 1\right )}^{2} {\left (x - 1\right )}^{7}\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)^7*(1+x)^2)^(1/3),x, algorithm="giac")

[Out]

integrate(((x + 1)^2*(x - 1)^7)^(-1/3), x)

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maple [A]  time = 0.05, size = 27, normalized size = 0.51

method	result	size
gosper	\(\frac {3 \left (1+x \right ) \left (-1+x \right ) \left (3 x -5\right )}{16 \left (\left (-1+x \right )^{7} \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\)	\(27\)
risch	\(\frac {3 \left (-1+x \right ) \left (3 x^{2}-2 x -5\right )}{16 \left (\left (-1+x \right )^{7} \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\)	\(29\)
trager	\(\frac {3 \left (3 x -5\right ) \left (x^{9}-5 x^{8}+8 x^{7}-14 x^{5}+14 x^{4}-8 x^{2}+5 x -1\right )^{\frac {2}{3}}}{16 \left (-1+x \right )^{6} \left (1+x \right )}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-1+x)^7*(1+x)^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/16*(1+x)*(-1+x)*(3*x-5)/((-1+x)^7*(1+x)^2)^(1/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left ({\left (x + 1\right )}^{2} {\left (x - 1\right )}^{7}\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)^7*(1+x)^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(((x + 1)^2*(x - 1)^7)^(-1/3), x)

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mupad [B]  time = 0.23, size = 30, normalized size = 0.57 \[ \frac {3\,\left (3\,x-5\right )\,{\left ({\left (x-1\right )}^7\,{\left (x+1\right )}^2\right )}^{2/3}}{16\,{\left (x-1\right )}^6\,\left (x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x - 1)^7*(x + 1)^2)^(1/3),x)

[Out]

(3*(3*x - 5)*((x - 1)^7*(x + 1)^2)^(2/3))/(16*(x - 1)^6*(x + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{\left (x - 1\right )^{7} \left (x + 1\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)**7*(1+x)**2)**(1/3),x)

[Out]

Integral(((x - 1)**7*(x + 1)**2)**(-1/3), x)

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