3.519 \(\int \frac{\sqrt [3]{a+b x^3}}{x^5} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^3\right )^{4/3}}{4 a x^4} \]

[Out]

-(a + b*x^3)^(4/3)/(4*a*x^4)

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Rubi [A]  time = 0.0044863, 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} \[ -\frac{\left (a+b x^3\right )^{4/3}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/x^5,x]

[Out]

-(a + b*x^3)^(4/3)/(4*a*x^4)

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{\sqrt [3]{a+b x^3}}{x^5} \, dx &=-\frac{\left (a+b x^3\right )^{4/3}}{4 a x^4}\\ \end{align*}

Mathematica [A]  time = 0.0040671, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^3\right )^{4/3}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/x^5,x]

[Out]

-(a + b*x^3)^(4/3)/(4*a*x^4)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a{x}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/x^5,x)

[Out]

-1/4*(b*x^3+a)^(4/3)/a/x^4

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Maxima [A]  time = 0.983614, size = 23, normalized size = 1.1 \begin{align*} -\frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}}}{4 \, a x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5,x, algorithm="maxima")

[Out]

-1/4*(b*x^3 + a)^(4/3)/(a*x^4)

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Fricas [A]  time = 1.94359, size = 43, normalized size = 2.05 \begin{align*} -\frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}}}{4 \, a x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5,x, algorithm="fricas")

[Out]

-1/4*(b*x^3 + a)^(4/3)/(a*x^4)

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Sympy [B]  time = 0.974972, size = 68, normalized size = 3.24 \begin{align*} \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{4}{3}\right )}{3 x^{3} \Gamma \left (- \frac{1}{3}\right )} + \frac{b^{\frac{4}{3}} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{4}{3}\right )}{3 a \Gamma \left (- \frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/x**5,x)

[Out]

b**(1/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-4/3)/(3*x**3*gamma(-1/3)) + b**(4/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-4/
3)/(3*a*gamma(-1/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/x^5, x)