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

Optimal. Leaf size=19 \[ -\frac{\sqrt [3]{a+b x^3}}{a x} \]

[Out]

-((a + b*x^3)^(1/3)/(a*x))

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Rubi [A]  time = 0.0048431, antiderivative size = 19, 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 = {264} \[ -\frac{\sqrt [3]{a+b x^3}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^3)^(2/3)),x]

[Out]

-((a + b*x^3)^(1/3)/(a*x))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b x^3\right )^{2/3}} \, dx &=-\frac{\sqrt [3]{a+b x^3}}{a x}\\ \end{align*}

Mathematica [A]  time = 0.003754, size = 19, normalized size = 1. \[ -\frac{\sqrt [3]{a+b x^3}}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x^3)^(1/3)/(a*x))

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Maple [A]  time = 0.004, size = 18, normalized size = 1. \begin{align*} -{\frac{1}{ax}\sqrt [3]{b{x}^{3}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01348, size = 23, normalized size = 1.21 \begin{align*} -\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x^3 + a)^(1/3)/(a*x)

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Fricas [A]  time = 1.50717, size = 35, normalized size = 1.84 \begin{align*} -\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x^3 + a)^(1/3)/(a*x)

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Sympy [B]  time = 0.612493, size = 31, normalized size = 1.63 \begin{align*} \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**(1/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-1/3)/(3*a*gamma(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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