3.300 \(\int \frac {x^6}{(1+x^7)^{5/3}} \, dx\)

Optimal. Leaf size=13 \[ -\frac {3}{14 \left (x^7+1\right )^{2/3}} \]

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Rubi [A]  time = 0.00, antiderivative size = 13, 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 = {261} \[ -\frac {3}{14 \left (x^7+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(1 + x^7)^(5/3),x]

[Out]

-3/(14*(1 + x^7)^(2/3))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x^6}{\left (1+x^7\right )^{5/3}} \, dx &=-\frac {3}{14 \left (1+x^7\right )^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {3}{14 \left (x^7+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(1 + x^7)^(5/3),x]

[Out]

-3/(14*(1 + x^7)^(2/3))

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IntegrateAlgebraic [A]  time = 0.03, size = 13, normalized size = 1.00 \[ -\frac {3}{14 \left (x^7+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6/(1 + x^7)^(5/3),x]

[Out]

-3/(14*(1 + x^7)^(2/3))

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fricas [A]  time = 0.58, size = 9, normalized size = 0.69 \[ -\frac {3}{14 \, {\left (x^{7} + 1\right )}^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^7+1)^(5/3),x, algorithm="fricas")

[Out]

-3/14/(x^7 + 1)^(2/3)

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giac [A]  time = 0.62, size = 9, normalized size = 0.69 \[ -\frac {3}{14 \, {\left (x^{7} + 1\right )}^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^7+1)^(5/3),x, algorithm="giac")

[Out]

-3/14/(x^7 + 1)^(2/3)

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maple [A]  time = 0.27, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(-\frac {3}{14 \left (x^{7}+1\right )^{\frac {2}{3}}}\)	\(10\)
default	\(-\frac {3}{14 \left (x^{7}+1\right )^{\frac {2}{3}}}\)	\(10\)
trager	\(-\frac {3}{14 \left (x^{7}+1\right )^{\frac {2}{3}}}\)	\(10\)
risch	\(-\frac {3}{14 \left (x^{7}+1\right )^{\frac {2}{3}}}\)	\(10\)
meijerg	\(\frac {x^{7} \hypergeom \left (\left [1, \frac {5}{3}\right ], \relax [2], -x^{7}\right )}{7}\)	\(17\)
gosper	\(-\frac {3 \left (1+x \right ) \left (x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x +1\right )}{14 \left (x^{7}+1\right )^{\frac {5}{3}}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^7+1)^(5/3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.60, size = 9, normalized size = 0.69 \[ -\frac {3}{14 \, {\left (x^{7} + 1\right )}^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^7+1)^(5/3),x, algorithm="maxima")

[Out]

-3/14/(x^7 + 1)^(2/3)

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mupad [B]  time = 0.32, size = 9, normalized size = 0.69 \[ -\frac {3}{14\,{\left (x^7+1\right )}^{2/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^7 + 1)^(5/3),x)

[Out]

-3/(14*(x^7 + 1)^(2/3))

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sympy [A]  time = 0.54, size = 12, normalized size = 0.92 \[ - \frac {3}{14 \left (x^{7} + 1\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(x**7+1)**(5/3),x)

[Out]

-3/(14*(x**7 + 1)**(2/3))

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