\(\int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx\) [273]

Optimal result

Integrand size = 15, antiderivative size = 25 \[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=\frac {3 \left (-2+3 x^4\right ) \left (x^2+x^6\right )^{2/3}}{40 x^8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2041, 2039} \[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=\frac {9 \left (x^6+x^2\right )^{2/3}}{40 x^4}-\frac {3 \left (x^6+x^2\right )^{2/3}}{20 x^8} \]

[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 = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=\frac {3 \left (-2+3 x^4\right ) \left (x^2+x^6\right )^{2/3}}{40 x^8} \]

[In]

Integrate[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)

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
meijerg	\(-\frac {3 \left (1-\frac {3 x^{4}}{2}\right ) \left (x^{4}+1\right )^{\frac {2}{3}}}{20 x^{\frac {20}{3}}}\)	\(20\)
trager	\(\frac {3 \left (3 x^{4}-2\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}}{40 x^{8}}\)	\(22\)
pseudoelliptic	\(\frac {3 \left (3 x^{4}-2\right ) \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {2}{3}}}{40 x^{8}}\)	\(24\)
gosper	\(\frac {3 \left (x^{4}+1\right ) \left (3 x^{4}-2\right )}{40 x^{6} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}}\)	\(27\)
risch	\(\frac {\frac {3}{40} x^{4}-\frac {3}{20}+\frac {9}{40} x^{8}}{x^{6} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}}}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=\frac {3 \, {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x^{4} - 2\right )}}{40 \, x^{8}} \]

[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]

\[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=\int \frac {1}{x^{7} \sqrt [3]{x^{2} \left (x^{4} + 1\right )}}\, dx \]

[In]

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

[Out]

Integral(1/(x**7*(x**2*(x**4 + 1))**(1/3)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=\frac {3 \, {\left (3 \, x^{10} + x^{6} - 2 \, x^{2}\right )}}{40 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {13}{3}}} \]

[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^4 + 1)^(1/3)*(x^2)^(13/3))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3}{20} \, {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {5}{3}} + \frac {3}{8} \, {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \]

[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 = 5.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^7 \sqrt [3]{x^2+x^6}} \, dx=-\frac {6\,{\left (x^6+x^2\right )}^{2/3}-9\,x^4\,{\left (x^6+x^2\right )}^{2/3}}{40\,x^8} \]

[In]

int(1/(x^7*(x^2 + x^6)^(1/3)),x)

[Out]

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