3.1.72 \(\int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx\)

Optimal. Leaf size=16 \[ -\frac {\left (x^3+1\right )^{4/3}}{4 x^4} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} -\frac {\left (x^3+1\right )^{4/3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^3)^(1/3)/x^5,x]

[Out]

-1/4*(1 + x^3)^(4/3)/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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\left (x^3+1\right )^{4/3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^3)^(1/3)/x^5,x]

[Out]

-1/4*(1 + x^3)^(4/3)/x^4

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IntegrateAlgebraic [A]  time = 0.07, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^3\right )^{4/3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^3)^(1/3)/x^5,x]

[Out]

-1/4*(1 + x^3)^(4/3)/x^4

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fricas [A]  time = 0.45, size = 12, normalized size = 0.75 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(1/3)/x^5,x, algorithm="fricas")

[Out]

-1/4*(x^3 + 1)^(4/3)/x^4

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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}}}{x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(1/3)/x^5,x, algorithm="giac")

[Out]

integrate((x^3 + 1)^(1/3)/x^5, x)

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maple [A]  time = 0.07, size = 13, normalized size = 0.81

method	result	size
trager	\(-\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
meijerg	\(-\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
risch	\(-\frac {x^{6}+2 x^{3}+1}{4 \left (x^{3}+1\right )^{\frac {2}{3}} x^{4}}\)	\(23\)
gosper	\(-\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{4 x^{4}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(1/3)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.55, size = 12, normalized size = 0.75 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(1/3)/x^5,x, algorithm="maxima")

[Out]

-1/4*(x^3 + 1)^(4/3)/x^4

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mupad [B]  time = 0.19, size = 24, normalized size = 1.50 \begin {gather*} -\frac {{\left (x^3+1\right )}^{1/3}+x^3\,{\left (x^3+1\right )}^{1/3}}{4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 + 1)^(1/3)/x^5,x)

[Out]

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

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sympy [B]  time = 0.60, size = 53, normalized size = 3.31 \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]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)**(1/3)/x**5,x)

[Out]

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

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