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

Optimal. Leaf size=28 \[ \frac {\sqrt [3]{x^3-1} \left (5 x^6-3 x^3-2\right )}{14 x^7} \]

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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^3-1\right )^{4/3}}{7 x^7}+\frac {5 \left (x^3-1\right )^{4/3}}{14 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)^(1/3)*(1 + x^3))/x^8,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (5 x^3+2\right )}{14 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)^(1/3)*(1 + x^3))/x^8,x]

[Out]

((-1 + x^3)^(4/3)*(2 + 5*x^3))/(14*x^7)

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IntegrateAlgebraic [A]  time = 0.09, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-1+x^3} \left (-2-3 x^3+5 x^6\right )}{14 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^8,x]

[Out]

((-1 + x^3)^(1/3)*(-2 - 3*x^3 + 5*x^6))/(14*x^7)

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fricas [A]  time = 0.47, size = 24, normalized size = 0.86 \begin {gather*} \frac {{\left (5 \, x^{6} - 3 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{14 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(5*x^6 - 3*x^3 - 2)*(x^3 - 1)^(1/3)/x^7

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 25, normalized size = 0.89

method	result	size
trager	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{6}-3 x^{3}-2\right )}{14 x^{7}}\)	\(25\)
risch	\(\frac {5 x^{9}-8 x^{6}+x^{3}+2}{14 \left (x^{3}-1\right )^{\frac {2}{3}} x^{7}}\)	\(28\)
gosper	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{3}+2\right ) \left (-1+x \right ) \left (x^{2}+x +1\right )}{14 x^{7}}\)	\(29\)
meijerg	\(-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x^{4}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{7 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x^{7}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(x^3-1)^(1/3)*(5*x^6-3*x^3-2)/x^7

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maxima [A]  time = 0.38, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{2 \, x^{4}} - \frac {{\left (x^{3} - 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.25, size = 39, normalized size = 1.39 \begin {gather*} -\frac {2\,{\left (x^3-1\right )}^{1/3}+3\,x^3\,{\left (x^3-1\right )}^{1/3}-5\,x^6\,{\left (x^3-1\right )}^{1/3}}{14\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(1/3)*(x^3 + 1))/x^8,x)

[Out]

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

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sympy [C]  time = 2.23, size = 416, normalized size = 14.86 \begin {gather*} \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 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 {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{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}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {3 x^{6} \sqrt [3]{1 - \frac {1}{x^{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}}} \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}}} \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}}} \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 {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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