\(\int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx\) [274]

Optimal result

Integrand size = 13, antiderivative size = 26 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-4+x^3+3 x^6\right )}{28 x^7} \]

[Out]

1/28*(x^3-1)^(1/3)*(3*x^6+x^3-4)/x^7

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 270} \[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\frac {\left (x^3-1\right )^{4/3}}{7 x^7}+\frac {3 \left (x^3-1\right )^{4/3}}{28 x^4} \]

[In]

Int[(-1 + x^3)^(1/3)/x^8,x]

[Out]

(-1 + x^3)^(4/3)/(7*x^7) + (3*(-1 + x^3)^(4/3))/(28*x^4)

Rule 270

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

Rule 277

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^3\right )^{4/3}}{7 x^7}+\frac {3}{7} \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx \\ & = \frac {\left (-1+x^3\right )^{4/3}}{7 x^7}+\frac {3 \left (-1+x^3\right )^{4/3}}{28 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-4+x^3+3 x^6\right )}{28 x^7} \]

[In]

Integrate[(-1 + x^3)^(1/3)/x^8,x]

[Out]

((-1 + x^3)^(1/3)*(-4 + x^3 + 3*x^6))/(28*x^7)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {\left (x^{3}-1\right )^{\frac {4}{3}} \left (3 x^{3}+4\right )}{28 x^{7}}\)	\(20\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (3 x^{6}+x^{3}-4\right )}{28 x^{7}}\)	\(23\)
gosper	\(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (3 x^{3}+4\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{28 x^{7}}\)	\(29\)
risch	\(\frac {3 x^{9}-2 x^{6}-5 x^{3}+4}{28 \left (x^{3}-1\right )^{\frac {2}{3}} x^{7}}\)	\(30\)
meijerg	\(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{7 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{7}}\)	\(45\)

[In]

int((x^3-1)^(1/3)/x^8,x,method=_RETURNVERBOSE)

[Out]

1/28*(x^3-1)^(4/3)*(3*x^3+4)/x^7

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\frac {{\left (3 \, x^{6} + x^{3} - 4\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \]

[In]

integrate((x^3-1)^(1/3)/x^8,x, algorithm="fricas")

[Out]

1/28*(3*x^6 + x^3 - 4)*(x^3 - 1)^(1/3)/x^7

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 296, normalized size of antiderivative = 11.38 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\begin {cases} \frac {3 x^{6} \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 x^{3} \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {5 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{x^{3} \cdot \left (9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \]

[In]

integrate((x**3-1)**(1/3)/x**8,x)

[Out]

Piecewise((3*x**6*(-1 + x**(-3))**(1/3)*exp(I*pi/3)*gamma(-7/3)/(9*x**6*gamma(-1/3) - 9*x**3*gamma(-1/3)) - 2*
x**3*(-1 + x**(-3))**(1/3)*exp(I*pi/3)*gamma(-7/3)/(9*x**6*gamma(-1/3) - 9*x**3*gamma(-1/3)) - 5*(-1 + x**(-3)
)**(1/3)*exp(I*pi/3)*gamma(-7/3)/(9*x**6*gamma(-1/3) - 9*x**3*gamma(-1/3)) + 4*(-1 + x**(-3))**(1/3)*exp(I*pi/
3)*gamma(-7/3)/(x**3*(9*x**6*gamma(-1/3) - 9*x**3*gamma(-1/3))), 1/Abs(x**3) > 1), ((1 - 1/x**3)**(1/3)*gamma(
-7/3)/(3*gamma(-1/3)) + (1 - 1/x**3)**(1/3)*gamma(-7/3)/(9*x**3*gamma(-1/3)) - 4*(1 - 1/x**3)**(1/3)*gamma(-7/
3)/(9*x**6*gamma(-1/3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} - \frac {{\left (x^{3} - 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} \]

[In]

integrate((x^3-1)^(1/3)/x^8,x, algorithm="maxima")

[Out]

1/4*(x^3 - 1)^(4/3)/x^4 - 1/7*(x^3 - 1)^(7/3)/x^7

Giac [F]

\[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{8}} \,d x } \]

[In]

integrate((x^3-1)^(1/3)/x^8,x, algorithm="giac")

[Out]

integrate((x^3 - 1)^(1/3)/x^8, x)

Mupad [B] (verification not implemented)

Time = 5.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx=\frac {x^3\,{\left (x^3-1\right )}^{1/3}-4\,{\left (x^3-1\right )}^{1/3}+3\,x^6\,{\left (x^3-1\right )}^{1/3}}{28\,x^7} \]

[In]

int((x^3 - 1)^(1/3)/x^8,x)

[Out]

(x^3*(x^3 - 1)^(1/3) - 4*(x^3 - 1)^(1/3) + 3*x^6*(x^3 - 1)^(1/3))/(28*x^7)