\(\int \frac {\sqrt [3]{-1+x^3} (1+x^3)}{x^{11}} \, dx\) [399]

Optimal result

Integrand size = 18, antiderivative size = 33 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-7-9 x^3+4 x^6+12 x^9\right )}{70 x^{10}} \]

[Out]

1/70*(x^3-1)^(1/3)*(12*x^9+4*x^6-9*x^3-7)/x^10

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {464, 277, 270} \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\frac {\left (x^3-1\right )^{4/3}}{10 x^{10}}+\frac {8 \left (x^3-1\right )^{4/3}}{35 x^7}+\frac {6 \left (x^3-1\right )^{4/3}}{35 x^4} \]

[In]

Int[((-1 + x^3)^(1/3)*(1 + x^3))/x^11,x]

[Out]

(-1 + x^3)^(4/3)/(10*x^10) + (8*(-1 + x^3)^(4/3))/(35*x^7) + (6*(-1 + x^3)^(4/3))/(35*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]

Rule 464

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {8}{5} \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx \\ & = \frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {8 \left (-1+x^3\right )^{4/3}}{35 x^7}+\frac {24}{35} \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx \\ & = \frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {8 \left (-1+x^3\right )^{4/3}}{35 x^7}+\frac {6 \left (-1+x^3\right )^{4/3}}{35 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\frac {\left (-1+x^3\right )^{4/3} \left (7+16 x^3+12 x^6\right )}{70 x^{10}} \]

[In]

Integrate[((-1 + x^3)^(1/3)*(1 + x^3))/x^11,x]

[Out]

((-1 + x^3)^(4/3)*(7 + 16*x^3 + 12*x^6))/(70*x^10)

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(\frac {\left (12 x^{6}+16 x^{3}+7\right ) \left (x^{3}-1\right )^{\frac {4}{3}}}{70 x^{10}}\)	\(25\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (12 x^{9}+4 x^{6}-9 x^{3}-7\right )}{70 x^{10}}\)	\(30\)
gosper	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (12 x^{6}+16 x^{3}+7\right ) \left (x -1\right ) \left (x^{2}+x +1\right )}{70 x^{10}}\)	\(34\)
risch	\(\frac {12 x^{12}-8 x^{9}-13 x^{6}+2 x^{3}+7}{70 \left (x^{3}-1\right )^{\frac {2}{3}} x^{10}}\)	\(35\)
meijerg	\(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{7 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{7}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {9}{14} x^{9}-\frac {3}{14} x^{6}-\frac {1}{7} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{10 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{10}}\)	\(95\)

[In]

int((x^3-1)^(1/3)*(x^3+1)/x^11,x,method=_RETURNVERBOSE)

[Out]

1/70*(12*x^6+16*x^3+7)/x^10*(x^3-1)^(4/3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\frac {{\left (12 \, x^{9} + 4 \, x^{6} - 9 \, x^{3} - 7\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{70 \, x^{10}} \]

[In]

integrate((x^3-1)^(1/3)*(x^3+1)/x^11,x, algorithm="fricas")

[Out]

1/70*(12*x^9 + 4*x^6 - 9*x^3 - 7)*(x^3 - 1)^(1/3)/x^10

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.42 (sec) , antiderivative size = 570, normalized size of antiderivative = 17.27 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\begin {cases} \frac {2 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {2 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {28 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{9} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {2 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {2 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{6} \Gamma \left (- \frac {1}{3}\right )} + \frac {28 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{9} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \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 {4 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{9} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{6} \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 )} & \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**3+1)/x**11,x)

[Out]

Piecewise((2*(-1 + x**(-3))**(1/3)*exp(-2*I*pi/3)*gamma(-10/3)/(3*gamma(-1/3)) + 2*(-1 + x**(-3))**(1/3)*exp(-
2*I*pi/3)*gamma(-10/3)/(9*x**3*gamma(-1/3)) + 4*(-1 + x**(-3))**(1/3)*exp(-2*I*pi/3)*gamma(-10/3)/(27*x**6*gam
ma(-1/3)) - 28*(-1 + x**(-3))**(1/3)*exp(-2*I*pi/3)*gamma(-10/3)/(27*x**9*gamma(-1/3)), 1/Abs(x**3) > 1), (-2*
(1 - 1/x**3)**(1/3)*gamma(-10/3)/(3*gamma(-1/3)) - 2*(1 - 1/x**3)**(1/3)*gamma(-10/3)/(9*x**3*gamma(-1/3)) - 4
*(1 - 1/x**3)**(1/3)*gamma(-10/3)/(27*x**6*gamma(-1/3)) + 28*(1 - 1/x**3)**(1/3)*gamma(-10/3)/(27*x**9*gamma(-
1/3)), True)) + 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*g
amma(-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)) +
4*(-1 + x**(-3))**(1/3)*exp(I*pi/3)*gamma(-7/3)/(9*x**9*gamma(-1/3) - 9*x**6*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)), 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 = 37, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{2 \, x^{4}} - \frac {3 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} + \frac {{\left (x^{3} - 1\right )}^{\frac {10}{3}}}{10 \, x^{10}} \]

[In]

integrate((x^3-1)^(1/3)*(x^3+1)/x^11,x, algorithm="maxima")

[Out]

1/2*(x^3 - 1)^(4/3)/x^4 - 3/7*(x^3 - 1)^(7/3)/x^7 + 1/10*(x^3 - 1)^(10/3)/x^10

Giac [F]

\[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{11}} \,d x } \]

[In]

integrate((x^3-1)^(1/3)*(x^3+1)/x^11,x, algorithm="giac")

[Out]

integrate((x^3 + 1)*(x^3 - 1)^(1/3)/x^11, x)

Mupad [B] (verification not implemented)

Time = 5.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx=\frac {6\,{\left (x^3-1\right )}^{1/3}}{35\,x}+\frac {2\,{\left (x^3-1\right )}^{1/3}}{35\,x^4}-\frac {9\,{\left (x^3-1\right )}^{1/3}}{70\,x^7}-\frac {{\left (x^3-1\right )}^{1/3}}{10\,x^{10}} \]

[In]

int(((x^3 - 1)^(1/3)*(x^3 + 1))/x^11,x)

[Out]

(6*(x^3 - 1)^(1/3))/(35*x) + (2*(x^3 - 1)^(1/3))/(35*x^4) - (9*(x^3 - 1)^(1/3))/(70*x^7) - (x^3 - 1)^(1/3)/(10
*x^10)