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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {a}{x}-\frac {3 b}{2 x^{2/3}} \]

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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fricas [A]  time = 0.62, size = 15, normalized size = 0.88 \[ -\frac {3 \, b x^{\frac {1}{3}} + 2 \, a}{2 \, x} \]

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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giac [A]  time = 0.19, size = 15, normalized size = 0.88 \[ -\frac {3 \, b x^{\frac {1}{3}} + 2 \, a}{2 \, x} \]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ -\frac {3 b}{2 x^{\frac {2}{3}}}-\frac {a}{x} \]

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 = 0.91, size = 15, normalized size = 0.88 \[ -\frac {3 \, b x^{\frac {1}{3}} + 2 \, a}{2 \, x} \]

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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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \[ -\frac {a}{x}-\frac {3\,b}{2\,x^{2/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))/x^2,x)

[Out]

- a/x - (3*b)/(2*x^(2/3))

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sympy [A]  time = 0.73, size = 14, normalized size = 0.82 \[ - \frac {a}{x} - \frac {3 b}{2 x^{\frac {2}{3}}} \]

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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