3.4.10 \(\int \frac {(1+x^3)^{2/3} (2+x^3)}{x^9} \, dx\)

Optimal. Leaf size=28 \[ \frac {\left (x^3+1\right )^{2/3} \left (-x^6-6 x^3-5\right )}{20 x^8} \]

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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 )^{5/3}}{4 x^8}-\frac {\left (x^3+1\right )^{5/3}}{20 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x^3)^(2/3)*(2 + x^3))/x^9,x]

[Out]

-1/4*(1 + x^3)^(5/3)/x^8 - (1 + x^3)^(5/3)/(20*x^5)

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 )^{2/3} \left (2+x^3\right )}{x^9} \, dx &=-\frac {\left (1+x^3\right )^{5/3}}{4 x^8}+\frac {1}{4} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (1+x^3\right )^{5/3}}{4 x^8}-\frac {\left (1+x^3\right )^{5/3}}{20 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.75 \begin {gather*} -\frac {\left (x^3+1\right )^{5/3} \left (x^3+5\right )}{20 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(2 + x^3))/x^9,x]

[Out]

-1/20*((1 + x^3)^(5/3)*(5 + x^3))/x^8

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IntegrateAlgebraic [A]  time = 0.09, size = 28, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (-5-6 x^3-x^6\right )}{20 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3))/x^9,x]

[Out]

((1 + x^3)^(2/3)*(-5 - 6*x^3 - x^6))/(20*x^8)

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fricas [A]  time = 0.43, size = 22, normalized size = 0.79 \begin {gather*} -\frac {{\left (x^{6} + 6 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{20 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)/x^9,x, algorithm="fricas")

[Out]

-1/20*(x^6 + 6*x^3 + 5)*(x^3 + 1)^(2/3)/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)/x^9,x, algorithm="giac")

[Out]

integrate((x^3 + 2)*(x^3 + 1)^(2/3)/x^9, x)

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maple [A]  time = 0.06, size = 23, normalized size = 0.82

method	result	size
trager	\(-\frac {\left (x^{6}+6 x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{20 x^{8}}\)	\(23\)
risch	\(-\frac {x^{9}+7 x^{6}+11 x^{3}+5}{20 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}\)	\(28\)
gosper	\(-\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{20 x^{8}}\)	\(29\)
meijerg	\(-\frac {\left (x^{3}+1\right )^{\frac {5}{3}}}{5 x^{5}}-\frac {\left (-\frac {3}{5} x^{6}+\frac {2}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{4 x^{8}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(x^3+2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/20*(x^6+6*x^3+5)/x^8*(x^3+1)^(2/3)

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maxima [A]  time = 0.65, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} + 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)/x^9,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.29, size = 22, normalized size = 0.79 \begin {gather*} -\frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6+6\,x^3+5\right )}{20\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + 1)^(2/3)*(x^3 + 2))/x^9,x)

[Out]

-((x^3 + 1)^(2/3)*(6*x^3 + x^6 + 5))/(20*x^8)

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sympy [B]  time = 2.38, size = 139, normalized size = 4.96 \begin {gather*} \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 x^{2} \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {4 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{5} \Gamma \left (- \frac {2}{3}\right )} - \frac {10 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{8} \Gamma \left (- \frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)**(2/3)*(x**3+2)/x**9,x)

[Out]

(1 + x**(-3))**(2/3)*gamma(-5/3)/(3*gamma(-2/3)) + 2*(x**3 + 1)**(2/3)*gamma(-8/3)/(3*x**2*gamma(-2/3)) + (1 +
x**(-3))**(2/3)*gamma(-5/3)/(3*x**3*gamma(-2/3)) - 4*(x**3 + 1)**(2/3)*gamma(-8/3)/(9*x**5*gamma(-2/3)) - 10*
(x**3 + 1)**(2/3)*gamma(-8/3)/(9*x**8*gamma(-2/3))

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