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

Optimal. Leaf size=33 \[ \frac {\sqrt [3]{x^3+1} \left (12 x^9-4 x^6-9 x^3+7\right )}{70 x^{10}} \]

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Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {453, 271, 264} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)*(1 + x^3)^(1/3))/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 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 271

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 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^{11}} \, dx &=\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*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.85 \begin {gather*} \frac {\left (x^3+1\right )^{4/3} \left (12 x^6-16 x^3+7\right )}{70 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.09, size = 28, normalized size = 0.85 \begin {gather*} \frac {\left (1+x^3\right )^{4/3} \left (7-16 x^3+12 x^6\right )}{70 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 29, normalized size = 0.88 \begin {gather*} \frac {{\left (12 \, x^{9} - 4 \, x^{6} - 9 \, x^{3} + 7\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{70 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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^{11}}\,{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^11,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 30, normalized size = 0.91

method	result	size
trager	\(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (12 x^{9}-4 x^{6}-9 x^{3}+7\right )}{70 x^{10}}\)	\(30\)
risch	\(\frac {12 x^{12}+8 x^{9}-13 x^{6}-2 x^{3}+7}{70 \left (x^{3}+1\right )^{\frac {2}{3}} x^{10}}\)	\(35\)
gosper	\(\frac {\left (x^{2}-x +1\right ) \left (1+x \right ) \left (12 x^{6}-16 x^{3}+7\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{70 x^{10}}\)	\(36\)
meijerg	\(\frac {\left (\frac {9}{14} x^{9}-\frac {3}{14} x^{6}+\frac {1}{7} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{10 x^{10}}-\frac {\left (-\frac {3}{4} x^{6}+\frac {1}{4} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{7 x^{7}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 37, normalized size = 1.12 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 0.33, size = 49, normalized size = 1.48 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)*(x^3 + 1)^(1/3))/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)

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sympy [B]  time = 2.70, size = 199, normalized size = 6.03 \begin {gather*} - \frac {2 \sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{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 {2 \sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{9 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]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{6} \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 )} - \frac {28 \sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{9} \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**11,x)

[Out]

-2*(1 + x**(-3))**(1/3)*gamma(-10/3)/(3*gamma(-1/3)) + (x**3 + 1)**(1/3)*gamma(-7/3)/(3*x*gamma(-1/3)) + 2*(1
+ x**(-3))**(1/3)*gamma(-10/3)/(9*x**3*gamma(-1/3)) - (x**3 + 1)**(1/3)*gamma(-7/3)/(9*x**4*gamma(-1/3)) - 4*(
1 + x**(-3))**(1/3)*gamma(-10/3)/(27*x**6*gamma(-1/3)) - 4*(x**3 + 1)**(1/3)*gamma(-7/3)/(9*x**7*gamma(-1/3))
- 28*(1 + x**(-3))**(1/3)*gamma(-10/3)/(27*x**9*gamma(-1/3))

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