3.2291 \(\int \frac{a+b \sqrt [3]{x}}{x^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{x}-\frac{3 b}{2 x^{2/3}} \]

[Out]

-(a/x) - (3*b)/(2*x^(2/3))

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Rubi [A]  time = 0.0050521, 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} \[ -\frac{a}{x}-\frac{3 b}{2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))/x^2,x]

[Out]

-(a/x) - (3*b)/(2*x^(2/3))

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^2} \, dx &=\int \left (\frac{a}{x^2}+\frac{b}{x^{5/3}}\right ) \, dx\\ &=-\frac{a}{x}-\frac{3 b}{2 x^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0062583, size = 17, normalized size = 1. \[ -\frac{a}{x}-\frac{3 b}{2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))/x^2,x]

[Out]

-(a/x) - (3*b)/(2*x^(2/3))

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Maple [A]  time = 0.004, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{x}}-{\frac{3\,b}{2}{x}^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.978657, size = 20, normalized size = 1.18 \begin{align*} -\frac{3 \, b x^{\frac{1}{3}} + 2 \, a}{2 \, x} \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]

-1/2*(3*b*x^(1/3) + 2*a)/x

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Fricas [A]  time = 1.40431, size = 38, normalized size = 2.24 \begin{align*} -\frac{3 \, b x^{\frac{1}{3}} + 2 \, a}{2 \, x} \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]

-1/2*(3*b*x^(1/3) + 2*a)/x

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Sympy [A]  time = 0.940901, size = 14, normalized size = 0.82 \begin{align*} - \frac{a}{x} - \frac{3 b}{2 x^{\frac{2}{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/(2*x**(2/3))

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Giac [A]  time = 1.10605, size = 20, normalized size = 1.18 \begin{align*} -\frac{3 \, b x^{\frac{1}{3}} + 2 \, a}{2 \, x} \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]

-1/2*(3*b*x^(1/3) + 2*a)/x