3.2293 \(\int \frac{a+b \sqrt [3]{x}}{x^4} \, dx\)

Optimal. Leaf size=19 \[ -\frac{a}{3 x^3}-\frac{3 b}{8 x^{8/3}} \]

[Out]

-a/(3*x^3) - (3*b)/(8*x^(8/3))

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Rubi [A]  time = 0.0052725, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ -\frac{a}{3 x^3}-\frac{3 b}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))/x^4,x]

[Out]

-a/(3*x^3) - (3*b)/(8*x^(8/3))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a+b \sqrt [3]{x}}{x^4} \, dx &=\int \left (\frac{a}{x^4}+\frac{b}{x^{11/3}}\right ) \, dx\\ &=-\frac{a}{3 x^3}-\frac{3 b}{8 x^{8/3}}\\ \end{align*}

Mathematica [A]  time = 0.0061057, size = 19, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{3 b}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))/x^4,x]

[Out]

-a/(3*x^3) - (3*b)/(8*x^(8/3))

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Maple [A]  time = 0.006, size = 14, normalized size = 0.7 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{3\,b}{8}{x}^{-{\frac{8}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a/x^3-3/8*b/x^(8/3)

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Maxima [A]  time = 0.960897, size = 20, normalized size = 1.05 \begin{align*} -\frac{9 \, b x^{\frac{1}{3}} + 8 \, a}{24 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(9*b*x^(1/3) + 8*a)/x^3

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Fricas [A]  time = 1.46816, size = 42, normalized size = 2.21 \begin{align*} -\frac{9 \, b x^{\frac{1}{3}} + 8 \, a}{24 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(9*b*x^(1/3) + 8*a)/x^3

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Sympy [A]  time = 2.99122, size = 17, normalized size = 0.89 \begin{align*} - \frac{a}{3 x^{3}} - \frac{3 b}{8 x^{\frac{8}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(3*x**3) - 3*b/(8*x**(8/3))

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Giac [A]  time = 1.11104, size = 20, normalized size = 1.05 \begin{align*} -\frac{9 \, b x^{\frac{1}{3}} + 8 \, a}{24 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(9*b*x^(1/3) + 8*a)/x^3