3.1987 \(\int \frac{1}{(a+\frac{b}{x^3})^2 x^4} \, dx\)

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

[Out]

1/(3*b*(a + b/x^3))

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Rubi [A]  time = 0.004098, 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} \[ \frac{1}{3 b \left (a+\frac{b}{x^3}\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)^2*x^4),x]

[Out]

1/(3*b*(a + b/x^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{1}{\left (a+\frac{b}{x^3}\right )^2 x^4} \, dx &=\frac{1}{3 b \left (a+\frac{b}{x^3}\right )}\\ \end{align*}

Mathematica [A]  time = 0.00499, size = 16, normalized size = 1. \[ -\frac{1}{3 a \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)^2*x^4),x]

[Out]

-1/(3*a*(b + a*x^3))

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a \left ( a{x}^{3}+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^3)^2/x^4,x)

[Out]

-1/3/a/(a*x^3+b)

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Maxima [A]  time = 0.972891, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{3 \,{\left (a + \frac{b}{x^{3}}\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^4,x, algorithm="maxima")

[Out]

1/3/((a + b/x^3)*b)

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Fricas [A]  time = 1.36826, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{3 \,{\left (a^{2} x^{3} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^4,x, algorithm="fricas")

[Out]

-1/3/(a^2*x^3 + a*b)

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Sympy [A]  time = 0.424857, size = 15, normalized size = 0.94 \begin{align*} - \frac{1}{3 a^{2} x^{3} + 3 a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**3)**2/x**4,x)

[Out]

-1/(3*a**2*x**3 + 3*a*b)

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Giac [A]  time = 1.17674, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{3 \,{\left (a x^{3} + b\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^4,x, algorithm="giac")

[Out]

-1/3/((a*x^3 + b)*a)