3.2040 \(\int \frac{1}{(a+\frac{b}{x^3})^{3/2} x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]

[Out]

2/(3*b*Sqrt[a + b/x^3])

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Rubi [A]  time = 0.0063507, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)^(3/2)*x^4),x]

[Out]

2/(3*b*Sqrt[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 )^{3/2} x^4} \, dx &=\frac{2}{3 b \sqrt{a+\frac{b}{x^3}}}\\ \end{align*}

Mathematica [A]  time = 0.0067539, size = 18, normalized size = 1. \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)^(3/2)*x^4),x]

[Out]

2/(3*b*Sqrt[a + b/x^3])

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Maple [A]  time = 0.003, size = 29, normalized size = 1.6 \begin{align*}{\frac{2\,a{x}^{3}+2\,b}{3\,b{x}^{3}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/x^3*(a*x^3+b)/b/((a*x^3+b)/x^3)^(3/2)

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Maxima [A]  time = 0.982481, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{3 \, \sqrt{a + \frac{b}{x^{3}}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/(sqrt(a + b/x^3)*b)

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Fricas [B]  time = 1.4839, size = 63, normalized size = 3.5 \begin{align*} \frac{2 \, x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3 \,{\left (a b x^{3} + b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^3*sqrt((a*x^3 + b)/x^3)/(a*b*x^3 + b^2)

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Sympy [A]  time = 2.84906, size = 27, normalized size = 1.5 \begin{align*} \begin{cases} \frac{2}{3 b \sqrt{a + \frac{b}{x^{3}}}} & \text{for}\: b \neq 0 \\- \frac{1}{3 a^{\frac{3}{2}} x^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2/(3*b*sqrt(a + b/x**3)), Ne(b, 0)), (-1/(3*a**(3/2)*x**3), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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