\(\int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx\) [311]

Optimal result

Integrand size = 13, antiderivative size = 28 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x+x^3\right )^{2/3} \left (5-6 x^2+9 x^4\right )}{80 x^6} \]

[Out]

-3/80*(x^3+x)^(2/3)*(9*x^4-6*x^2+5)/x^6

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2041, 2039} \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x^3+x\right )^{2/3}}{16 x^6}+\frac {9 \left (x^3+x\right )^{2/3}}{40 x^4}-\frac {27 \left (x^3+x\right )^{2/3}}{80 x^2} \]

[In]

Int[1/(x^6*(x + x^3)^(1/3)),x]

[Out]

(-3*(x + x^3)^(2/3))/(16*x^6) + (9*(x + x^3)^(2/3))/(40*x^4) - (27*(x + x^3)^(2/3))/(80*x^2)

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+x^3\right )^{2/3}}{16 x^6}-\frac {3}{4} \int \frac {1}{x^4 \sqrt [3]{x+x^3}} \, dx \\ & = -\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (x+x^3\right )^{2/3}}{40 x^4}+\frac {9}{20} \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx \\ & = -\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (x+x^3\right )^{2/3}}{40 x^4}-\frac {27 \left (x+x^3\right )^{2/3}}{80 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x+x^3\right )^{2/3} \left (5-6 x^2+9 x^4\right )}{80 x^6} \]

[In]

Integrate[1/(x^6*(x + x^3)^(1/3)),x]

[Out]

(-3*(x + x^3)^(2/3)*(5 - 6*x^2 + 9*x^4))/(80*x^6)

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
trager	\(-\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}} \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{6}}\)	\(25\)
meijerg	\(-\frac {3 \left (\frac {9}{5} x^{4}-\frac {6}{5} x^{2}+1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}}{16 x^{\frac {16}{3}}}\)	\(25\)
pseudoelliptic	\(-\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {2}{3}} \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{6}}\)	\(27\)
gosper	\(-\frac {3 \left (x^{2}+1\right ) \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{5} \left (x^{3}+x \right )^{\frac {1}{3}}}\)	\(30\)
risch	\(-\frac {3 \left (9 x^{6}+3 x^{4}-x^{2}+5\right )}{80 x^{5} {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}\)	\(32\)

[In]

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

[Out]

-3/80*(x^3+x)^(2/3)*(9*x^4-6*x^2+5)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \, {\left (9 \, x^{4} - 6 \, x^{2} + 5\right )} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{80 \, x^{6}} \]

[In]

integrate(1/x^6/(x^3+x)^(1/3),x, algorithm="fricas")

[Out]

-3/80*(9*x^4 - 6*x^2 + 5)*(x^3 + x)^(2/3)/x^6

Sympy [F]

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

[In]

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

[Out]

Integral(1/(x**6*(x*(x**2 + 1))**(1/3)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \, {\left (9 \, x^{7} + 3 \, x^{5} - x^{3} + 5 \, x\right )}}{80 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{\frac {19}{3}}} \]

[In]

integrate(1/x^6/(x^3+x)^(1/3),x, algorithm="maxima")

[Out]

-3/80*(9*x^7 + 3*x^5 - x^3 + 5*x)/((x^2 + 1)^(1/3)*x^(19/3))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3}{16} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {8}{3}} + \frac {3}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} - \frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \]

[In]

integrate(1/x^6/(x^3+x)^(1/3),x, algorithm="giac")

[Out]

-3/16*(1/x^2 + 1)^(8/3) + 3/5*(1/x^2 + 1)^(5/3) - 3/4*(1/x^2 + 1)^(2/3)

Mupad [B] (verification not implemented)

Time = 5.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3\,{\left (x^3+x\right )}^{2/3}\,\left (9\,x^4-6\,x^2+5\right )}{80\,x^6} \]

[In]

int(1/(x^6*(x + x^3)^(1/3)),x)

[Out]

-(3*(x + x^3)^(2/3)*(9*x^4 - 6*x^2 + 5))/(80*x^6)