3.1334 \(\int \frac{x^5}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{6 b \left (a+b x^6\right )} \]

[Out]

-1/(6*b*(a + b*x^6))

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Rubi [A]  time = 0.0033566, antiderivative size = 16, 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} \[ -\frac{1}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^6)^2,x]

[Out]

-1/(6*b*(a + b*x^6))

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^5}{\left (a+b x^6\right )^2} \, dx &=-\frac{1}{6 b \left (a+b x^6\right )}\\ \end{align*}

Mathematica [A]  time = 0.0035409, size = 16, normalized size = 1. \[ -\frac{1}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^6)^2,x]

[Out]

-1/(6*b*(a + b*x^6))

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,b \left ( b{x}^{6}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^6+a)^2,x)

[Out]

-1/6/b/(b*x^6+a)

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Maxima [A]  time = 0.985081, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{6 \,{\left (b x^{6} + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

-1/6/((b*x^6 + a)*b)

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Fricas [A]  time = 1.36145, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{6 \,{\left (b^{2} x^{6} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^6+a)^2,x, algorithm="fricas")

[Out]

-1/6/(b^2*x^6 + a*b)

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Sympy [A]  time = 0.864557, size = 15, normalized size = 0.94 \begin{align*} - \frac{1}{6 a b + 6 b^{2} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**6+a)**2,x)

[Out]

-1/(6*a*b + 6*b**2*x**6)

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Giac [A]  time = 1.23156, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{6 \,{\left (b x^{6} + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^6+a)^2,x, algorithm="giac")

[Out]

-1/6/((b*x^6 + a)*b)