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

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

[Out]

-3/(14*(1 + x^7)^(2/3))

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Rubi [A]  time = 0.0029647, antiderivative size = 13, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0026527, size = 13, normalized size = 1. \[ -\frac{3}{14 \left (x^7+1\right )^{2/3}} \]

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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Maple [B]  time = 0.004, size = 37, normalized size = 2.9 \begin{align*} -{\frac{ \left ( 3+3\,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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/14*(1+x)*(x^6-x^5+x^4-x^3+x^2-x+1)/(x^7+1)^(5/3)

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Maxima [A]  time = 0.942658, size = 12, normalized size = 0.92 \begin{align*} -\frac{3}{14 \,{\left (x^{7} + 1\right )}^{\frac{2}{3}}} \end{align*}

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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Fricas [A]  time = 1.77936, size = 31, normalized size = 2.38 \begin{align*} -\frac{3}{14 \,{\left (x^{7} + 1\right )}^{\frac{2}{3}}} \end{align*}

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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Sympy [A]  time = 0.572686, size = 12, normalized size = 0.92 \begin{align*} - \frac{3}{14 \left (x^{7} + 1\right )^{\frac{2}{3}}} \end{align*}

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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Giac [A]  time = 1.06213, size = 12, normalized size = 0.92 \begin{align*} -\frac{3}{14 \,{\left (x^{7} + 1\right )}^{\frac{2}{3}}} \end{align*}

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)