3.2292 \(\int \frac{a+b \sqrt [3]{x}}{x^3} \, dx\)

Optimal. Leaf size=19 \[ -\frac{a}{2 x^2}-\frac{3 b}{5 x^{5/3}} \]

[Out]

-a/(2*x^2) - (3*b)/(5*x^(5/3))

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Rubi [A]  time = 0.0050065, 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} \[ -\frac{a}{2 x^2}-\frac{3 b}{5 x^{5/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))/x^3,x]

[Out]

-a/(2*x^2) - (3*b)/(5*x^(5/3))

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^3} \, dx &=\int \left (\frac{a}{x^3}+\frac{b}{x^{8/3}}\right ) \, dx\\ &=-\frac{a}{2 x^2}-\frac{3 b}{5 x^{5/3}}\\ \end{align*}

Mathematica [A]  time = 0.0063393, size = 19, normalized size = 1. \[ -\frac{a}{2 x^2}-\frac{3 b}{5 x^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))/x^3,x]

[Out]

-a/(2*x^2) - (3*b)/(5*x^(5/3))

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Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{3\,b}{5}{x}^{-{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/x^2*a-3/5*b/x^(5/3)

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Maxima [A]  time = 0.952807, size = 20, normalized size = 1.05 \begin{align*} -\frac{6 \, b x^{\frac{1}{3}} + 5 \, a}{10 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(6*b*x^(1/3) + 5*a)/x^2

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Fricas [A]  time = 1.43734, size = 42, normalized size = 2.21 \begin{align*} -\frac{6 \, b x^{\frac{1}{3}} + 5 \, a}{10 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(6*b*x^(1/3) + 5*a)/x^2

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Sympy [A]  time = 1.67627, size = 17, normalized size = 0.89 \begin{align*} - \frac{a}{2 x^{2}} - \frac{3 b}{5 x^{\frac{5}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(2*x**2) - 3*b/(5*x**(5/3))

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Giac [A]  time = 1.102, size = 20, normalized size = 1.05 \begin{align*} -\frac{6 \, b x^{\frac{1}{3}} + 5 \, a}{10 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(6*b*x^(1/3) + 5*a)/x^2