\(\int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx\) [304]

Optimal result

Integrand size = 17, antiderivative size = 27 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (2+3 x^4\right ) \left (-x^2+x^6\right )^{2/3}}{40 x^8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (x^6-x^2\right )^{2/3}}{20 x^8}+\frac {9 \left (x^6-x^2\right )^{2/3}}{40 x^4} \]

[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 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 10.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (x^2 \left (-1+x^4\right )\right )^{2/3} \left (2+3 x^4\right )}{40 x^8} \]

[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)

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x^{4} + 2\right )}}{40 \, x^{8}} \]

[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

Sympy [F]

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

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \, {\left (3 \, x^{10} - x^{6} - 2 \, x^{2}\right )}}{40 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {13}{3}}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=-\frac {3}{20} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {5}{3}} + \frac {3}{8} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \]

[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)

Mupad [B] (verification not implemented)

Time = 6.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {9\,x^4\,{\left (x^6-x^2\right )}^{2/3}+6\,{\left (x^6-x^2\right )}^{2/3}}{40\,x^8} \]

[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)