3.4.4 \(\int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx\)

Optimal. Leaf size=27 \[ \frac {3 \left (3 x^4+2\right ) \left (x^6-x^2\right )^{2/3}}{40 x^8} \]

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Rubi [A]  time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2016, 2014} \begin {gather*} \frac {3 \left (x^6-x^2\right )^{2/3}}{20 x^8}+\frac {9 \left (x^6-x^2\right )^{2/3}}{40 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(-x^2 + x^6)^(1/3)),x]

[Out]

(3*(-x^2 + x^6)^(2/3))/(20*x^8) + (9*(-x^2 + x^6)^(2/3))/(40*x^4)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx &=\frac {3 \left (-x^2+x^6\right )^{2/3}}{20 x^8}+\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx\\ &=\frac {3 \left (-x^2+x^6\right )^{2/3}}{20 x^8}+\frac {9 \left (-x^2+x^6\right )^{2/3}}{40 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^2 \left (x^4-1\right )\right )^{2/3} \left (3 x^4+2\right )}{40 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(-x^2 + x^6)^(1/3)),x]

[Out]

(3*(x^2*(-1 + x^4))^(2/3)*(2 + 3*x^4))/(40*x^8)

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IntegrateAlgebraic [A]  time = 0.42, size = 27, normalized size = 1.00 \begin {gather*} \frac {3 \left (2+3 x^4\right ) \left (-x^2+x^6\right )^{2/3}}{40 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^7*(-x^2 + x^6)^(1/3)),x]

[Out]

(3*(2 + 3*x^4)*(-x^2 + x^6)^(2/3))/(40*x^8)

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fricas [A]  time = 0.48, size = 23, normalized size = 0.85 \begin {gather*} \frac {3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x^{4} + 2\right )}}{40 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6-x^2)^(1/3),x, algorithm="fricas")

[Out]

3/40*(x^6 - x^2)^(2/3)*(3*x^4 + 2)/x^8

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giac [A]  time = 0.33, size = 23, normalized size = 0.85 \begin {gather*} -\frac {3}{20} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {5}{3}} + \frac {3}{8} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6-x^2)^(1/3),x, algorithm="giac")

[Out]

-3/20*(-1/x^4 + 1)^(5/3) + 3/8*(-1/x^4 + 1)^(2/3)

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maple [A]  time = 0.08, size = 24, normalized size = 0.89

method	result	size
trager	\(\frac {3 \left (3 x^{4}+2\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}}{40 x^{8}}\)	\(24\)
risch	\(\frac {-\frac {3}{40} x^{4}-\frac {3}{20}+\frac {9}{40} x^{8}}{x^{6} \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{3}}}\)	\(29\)
gosper	\(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (3 x^{4}+2\right )}{40 x^{6} \left (x^{6}-x^{2}\right )^{\frac {1}{3}}}\)	\(35\)
meijerg	\(-\frac {3 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \left (1+\frac {3 x^{4}}{2}\right ) \left (-x^{4}+1\right )^{\frac {2}{3}}}{20 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}} x^{\frac {20}{3}}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^6-x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/40*(3*x^4+2)*(x^6-x^2)^(2/3)/x^8

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maxima [A]  time = 0.48, size = 37, normalized size = 1.37 \begin {gather*} \frac {3 \, {\left (3 \, x^{10} - x^{6} - 2 \, x^{2}\right )}}{40 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {13}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6-x^2)^(1/3),x, algorithm="maxima")

[Out]

3/40*(3*x^10 - x^6 - 2*x^2)/((x^2 + 1)^(1/3)*(x^2 - 1)^(1/3)*(x^2)^(13/3))

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mupad [B]  time = 0.24, size = 35, normalized size = 1.30 \begin {gather*} \frac {9\,x^4\,{\left (x^6-x^2\right )}^{2/3}+6\,{\left (x^6-x^2\right )}^{2/3}}{40\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(x^6 - x^2)^(1/3)),x)

[Out]

(9*x^4*(x^6 - x^2)^(2/3) + 6*(x^6 - x^2)^(2/3))/(40*x^8)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**6-x**2)**(1/3),x)

[Out]

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

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