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

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

[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))

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Rubi [A]  time = 0.0198843, antiderivative size = 63, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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]

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 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]

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*}

Mathematica [A]  time = 0.0155586, 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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Maple [A]  time = 0.001, size = 27, normalized size = 0.5 \begin{align*}{\frac{ \left ( 3+3\,x \right ) \left ( -1+x \right ) \left ( 3\,x-5 \right ) }{16}{\frac{1}{\sqrt [3]{ \left ( -1+x \right ) ^{7} \left ( 1+x \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left ({\left (x + 1\right )}^{2}{\left (x - 1\right )}^{7}\right )^{\frac{1}{3}}}\,{d x} \end{align*}

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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Fricas [A]  time = 1.67343, size = 184, normalized size = 3.47 \begin{align*} \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 )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{\left (x - 1\right )^{7} \left (x + 1\right )^{2}}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left ({\left (x + 1\right )}^{2}{\left (x - 1\right )}^{7}\right )^{\frac{1}{3}}}\,{d x} \end{align*}

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)