3.2400 \(\int \frac{a+\frac{b}{\sqrt [3]{x}}}{x^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{x}-\frac{3 b}{4 x^{4/3}} \]

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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Rubi [A]  time = 0.0049935, antiderivative size = 17, 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} \[ -\frac{a}{x}-\frac{3 b}{4 x^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))/x^2,x]

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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+\frac{b}{\sqrt [3]{x}}}{x^2} \, dx &=\int \left (\frac{b}{x^{7/3}}+\frac{a}{x^2}\right ) \, dx\\ &=-\frac{3 b}{4 x^{4/3}}-\frac{a}{x}\\ \end{align*}

Mathematica [A]  time = 0.0074651, size = 17, normalized size = 1. \[ -\frac{a}{x}-\frac{3 b}{4 x^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))/x^2,x]

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\frac{3\,b}{4}{x}^{-{\frac{4}{3}}}}-{\frac{a}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04429, size = 63, normalized size = 3.71 \begin{align*} -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4}}{4 \, b^{3}} + \frac{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a}{b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{2}}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44508, size = 43, normalized size = 2.53 \begin{align*} -\frac{4 \, a x + 3 \, b x^{\frac{2}{3}}}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*a*x + 3*b*x^(2/3))/x^2

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Sympy [A]  time = 0.674316, size = 14, normalized size = 0.82 \begin{align*} - \frac{a}{x} - \frac{3 b}{4 x^{\frac{4}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x - 3*b/(4*x**(4/3))

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Giac [A]  time = 1.12303, size = 20, normalized size = 1.18 \begin{align*} -\frac{4 \, a x^{\frac{1}{3}} + 3 \, b}{4 \, x^{\frac{4}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*a*x^(1/3) + 3*b)/x^(4/3)