\(\int \frac {\sqrt [3]{-1+x^6} (-1+2 x^6)}{x^{15}} \, dx\) [337]

Optimal result

Integrand size = 20, antiderivative size = 28 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {\sqrt [3]{-1+x^6} \left (4-15 x^6+11 x^{12}\right )}{56 x^{14}} \]

[Out]

1/56*(x^6-1)^(1/3)*(11*x^12-15*x^6+4)/x^14

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {464, 270} \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {11 \left (x^6-1\right )^{4/3}}{56 x^8}-\frac {\left (x^6-1\right )^{4/3}}{14 x^{14}} \]

[In]

Int[((-1 + x^6)^(1/3)*(-1 + 2*x^6))/x^15,x]

[Out]

-1/14*(-1 + x^6)^(4/3)/x^14 + (11*(-1 + x^6)^(4/3))/(56*x^8)

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 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^6\right )^{4/3}}{14 x^{14}}+\frac {11}{7} \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx \\ & = -\frac {\left (-1+x^6\right )^{4/3}}{14 x^{14}}+\frac {11 \left (-1+x^6\right )^{4/3}}{56 x^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {\left (-1+x^6\right )^{4/3} \left (-4+11 x^6\right )}{56 x^{14}} \]

[In]

Integrate[((-1 + x^6)^(1/3)*(-1 + 2*x^6))/x^15,x]

[Out]

((-1 + x^6)^(4/3)*(-4 + 11*x^6))/(56*x^14)

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(\frac {\left (x^{6}-1\right )^{\frac {4}{3}} \left (11 x^{6}-4\right )}{56 x^{14}}\)	\(20\)
trager	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (11 x^{12}-15 x^{6}+4\right )}{56 x^{14}}\)	\(25\)
risch	\(\frac {11 x^{18}-26 x^{12}+19 x^{6}-4}{56 \left (x^{6}-1\right )^{\frac {2}{3}} x^{14}}\)	\(30\)
gosper	\(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (11 x^{6}-4\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{56 x^{14}}\)	\(40\)
meijerg	\(-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{8}}+\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{12}-\frac {1}{4} x^{6}+1\right ) \left (-x^{6}+1\right )^{\frac {1}{3}}}{14 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{14}}\)	\(78\)

[In]

int((x^6-1)^(1/3)*(2*x^6-1)/x^15,x,method=_RETURNVERBOSE)

[Out]

1/56*(x^6-1)^(4/3)*(11*x^6-4)/x^14

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {{\left (11 \, x^{12} - 15 \, x^{6} + 4\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{56 \, x^{14}} \]

[In]

integrate((x^6-1)^(1/3)*(2*x^6-1)/x^15,x, algorithm="fricas")

[Out]

1/56*(11*x^12 - 15*x^6 + 4)*(x^6 - 1)^(1/3)/x^14

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 15.18 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=2 \left (\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases}\right ) - \begin {cases} \frac {3 x^{12} \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{12} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 x^{6} \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{12} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{6} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{18} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{12} \Gamma \left (- \frac {1}{3}\right )} - \frac {5 \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{12} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {7}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 \sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{12} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \]

[In]

integrate((x**6-1)**(1/3)*(2*x**6-1)/x**15,x)

[Out]

2*Piecewise(((-1 + x**(-6))**(1/3)*exp(-2*I*pi/3)*gamma(-4/3)/(6*gamma(-1/3)) - (-1 + x**(-6))**(1/3)*exp(-2*I
*pi/3)*gamma(-4/3)/(6*x**6*gamma(-1/3)), 1/Abs(x**6) > 1), (-(1 - 1/x**6)**(1/3)*gamma(-4/3)/(6*gamma(-1/3)) +
(1 - 1/x**6)**(1/3)*gamma(-4/3)/(6*x**6*gamma(-1/3)), True)) - Piecewise((3*x**12*(-1 + x**(-6))**(1/3)*exp(I
*pi/3)*gamma(-7/3)/(18*x**12*gamma(-1/3) - 18*x**6*gamma(-1/3)) - 2*x**6*(-1 + x**(-6))**(1/3)*exp(I*pi/3)*gam
ma(-7/3)/(18*x**12*gamma(-1/3) - 18*x**6*gamma(-1/3)) + 4*(-1 + x**(-6))**(1/3)*exp(I*pi/3)*gamma(-7/3)/(18*x*
*18*gamma(-1/3) - 18*x**12*gamma(-1/3)) - 5*(-1 + x**(-6))**(1/3)*exp(I*pi/3)*gamma(-7/3)/(18*x**12*gamma(-1/3
) - 18*x**6*gamma(-1/3)), 1/Abs(x**6) > 1), ((1 - 1/x**6)**(1/3)*gamma(-7/3)/(6*gamma(-1/3)) + (1 - 1/x**6)**(
1/3)*gamma(-7/3)/(18*x**6*gamma(-1/3)) - 2*(1 - 1/x**6)**(1/3)*gamma(-7/3)/(9*x**12*gamma(-1/3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} + \frac {{\left (x^{6} - 1\right )}^{\frac {7}{3}}}{14 \, x^{14}} \]

[In]

integrate((x^6-1)^(1/3)*(2*x^6-1)/x^15,x, algorithm="maxima")

[Out]

1/8*(x^6 - 1)^(4/3)/x^8 + 1/14*(x^6 - 1)^(7/3)/x^14

Giac [F]

\[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\int { \frac {{\left (2 \, x^{6} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{15}} \,d x } \]

[In]

integrate((x^6-1)^(1/3)*(2*x^6-1)/x^15,x, algorithm="giac")

[Out]

integrate((2*x^6 - 1)*(x^6 - 1)^(1/3)/x^15, x)

Mupad [B] (verification not implemented)

Time = 5.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {7\,{\left (x^6-1\right )}^{4/3}+11\,{\left (x^6-1\right )}^{7/3}}{56\,x^{14}} \]

[In]

int(((x^6 - 1)^(1/3)*(2*x^6 - 1))/x^15,x)

[Out]

(7*(x^6 - 1)^(4/3) + 11*(x^6 - 1)^(7/3))/(56*x^14)