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

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 + x^3)^(1/3))/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 {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{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 (2-5 x^3\right ) \left (x^3+1\right )^{4/3}}{14 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.09, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (2-5 x^3\right ) \left (1+x^3\right )^{4/3}}{14 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, 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)*(x^3+1)^(1/3)/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 )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 25, normalized size = 0.89

method	result	size
trager	\(-\frac {\left (5 x^{6}+3 x^{3}-2\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{14 x^{7}}\)	\(25\)
risch	\(-\frac {5 x^{9}+8 x^{6}+x^{3}-2}{14 \left (x^{3}+1\right )^{\frac {2}{3}} x^{7}}\)	\(28\)
gosper	\(-\frac {\left (x^{2}-x +1\right ) \left (1+x \right ) \left (5 x^{3}-2\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{14 x^{7}}\)	\(31\)
meijerg	\(\frac {\left (-\frac {3}{4} x^{6}+\frac {1}{4} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{7 x^{7}}-\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, 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)*(x^3+1)^(1/3)/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 = 0.16, size = 39, normalized size = 1.39 \begin {gather*} -\frac {3\,x^3\,{\left (x^3+1\right )}^{1/3}-2\,{\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)*(x^3 + 1)^(1/3))/x^8,x)

[Out]

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

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sympy [B]  time = 2.14, size = 134, normalized size = 4.79 \begin {gather*} \frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{3 x \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 )} + \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{7} \Gamma \left (- \frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 + x**(-3))**(1/3)*gamma(-4/3)/(3*gamma(-1/3)) - (x**3 + 1)**(1/3)*gamma(-7/3)/(3*x*gamma(-1/3)) + (1 + x**(
-3))**(1/3)*gamma(-4/3)/(3*x**3*gamma(-1/3)) + (x**3 + 1)**(1/3)*gamma(-7/3)/(9*x**4*gamma(-1/3)) + 4*(x**3 +
1)**(1/3)*gamma(-7/3)/(9*x**7*gamma(-1/3))

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