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

Optimal. Leaf size=28 \[ \frac {\sqrt [3]{x^6-1} \left (5 x^{12}-3 x^6-2\right )}{28 x^{14}} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{14 x^{14}}+\frac {5 \left (x^6-1\right )^{4/3}}{28 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + x^6)^(4/3)/(14*x^14) + (5*(-1 + x^6)^(4/3))/(28*x^8)

Rule 264

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 453

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*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^{15}} \, dx &=\frac {\left (-1+x^6\right )^{4/3}}{14 x^{14}}+\frac {10}{7} \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx\\ &=\frac {\left (-1+x^6\right )^{4/3}}{14 x^{14}}+\frac {5 \left (-1+x^6\right )^{4/3}}{28 x^8}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (x^6-1\right )^{4/3} \left (5 x^6+2\right )}{28 x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x^6)^(4/3)*(2 + 5*x^6))/(28*x^14)

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IntegrateAlgebraic [A]  time = 0.53, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (-1+x^6\right )^{4/3} \left (2+5 x^6\right )}{28 x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x^6)^(4/3)*(2 + 5*x^6))/(28*x^14)

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fricas [A]  time = 0.49, size = 24, normalized size = 0.86 \begin {gather*} \frac {{\left (5 \, x^{12} - 3 \, x^{6} - 2\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{28 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(5*x^12 - 3*x^6 - 2)*(x^6 - 1)^(1/3)/x^14

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{15}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 25, normalized size = 0.89

method	result	size
trager	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (5 x^{12}-3 x^{6}-2\right )}{28 x^{14}}\)	\(25\)
risch	\(\frac {5 x^{18}-8 x^{12}+x^{6}+2}{28 \left (x^{6}-1\right )^{\frac {2}{3}} x^{14}}\)	\(28\)
gosper	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (5 x^{6}+2\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{28 x^{14}}\)	\(40\)
meijerg	\(-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{8 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{8}}-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{12}-\frac {1}{4} x^{6}+1\right ) \left (-x^{6}+1\right )^{\frac {1}{3}}}{14 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{14}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(x^6-1)^(1/3)*(5*x^12-3*x^6-2)/x^14

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maxima [A]  time = 0.44, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{4 \, x^{8}} - \frac {{\left (x^{6} - 1\right )}^{\frac {7}{3}}}{14 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.34, size = 24, normalized size = 0.86 \begin {gather*} \frac {7\,{\left (x^6-1\right )}^{4/3}+5\,{\left (x^6-1\right )}^{7/3}}{28\,x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*(x^6 - 1)^(4/3) + 5*(x^6 - 1)^(7/3))/(28*x^14)

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sympy [C]  time = 4.28, size = 416, normalized size = 14.86 \begin {gather*} \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} + \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{12} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {3 x^{12} \sqrt [3]{1 - \frac {1}{x^{6}}} \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}}} \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}}} \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}}} \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 {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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