3.557 \(\int \frac{1}{x^3 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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Rubi [A]  time = 0.0048383, antiderivative size = 21, 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} \[ -\frac{\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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

Mathematica [A]  time = 0.0036308, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.97992, size = 23, normalized size = 1.1 \begin{align*} -\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*x^3 + a)^(2/3)/(a*x^2)

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Fricas [A]  time = 1.51147, size = 43, normalized size = 2.05 \begin{align*} -\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*x^3 + a)^(2/3)/(a*x^2)

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Sympy [A]  time = 0.653812, size = 31, normalized size = 1.48 \begin{align*} \frac{b^{\frac{2}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{2}{3}\right )}{3 a \Gamma \left (\frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{1}{3}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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