3.2401 \(\int \frac{a+\frac{b}{\sqrt [3]{x}}}{x^3} \, dx\)

Optimal. Leaf size=19 \[ -\frac{a}{2 x^2}-\frac{3 b}{7 x^{7/3}} \]

[Out]

(-3*b)/(7*x^(7/3)) - a/(2*x^2)

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Rubi [A]  time = 0.0050827, 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} \[ -\frac{a}{2 x^2}-\frac{3 b}{7 x^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))/x^3,x]

[Out]

(-3*b)/(7*x^(7/3)) - a/(2*x^2)

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^3} \, dx &=\int \left (\frac{b}{x^{10/3}}+\frac{a}{x^3}\right ) \, dx\\ &=-\frac{3 b}{7 x^{7/3}}-\frac{a}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0078449, size = 19, normalized size = 1. \[ -\frac{a}{2 x^2}-\frac{3 b}{7 x^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))/x^3,x]

[Out]

(-3*b)/(7*x^(7/3)) - a/(2*x^2)

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Maple [A]  time = 0.005, size = 14, normalized size = 0.7 \begin{align*} -{\frac{3\,b}{7}{x}^{-{\frac{7}{3}}}}-{\frac{a}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/7*b/x^(7/3)-1/2/x^2*a

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Maxima [B]  time = 0.954495, size = 132, normalized size = 6.95 \begin{align*} -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7}}{7 \, b^{6}} + \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a}{2 \, b^{6}} - \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{2}}{b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{3}}{2 \, b^{6}} - \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{4}}{b^{6}} + \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{5}}{2 \, b^{6}} \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]

-3/7*(a + b/x^(1/3))^7/b^6 + 5/2*(a + b/x^(1/3))^6*a/b^6 - 6*(a + b/x^(1/3))^5*a^2/b^6 + 15/2*(a + b/x^(1/3))^
4*a^3/b^6 - 5*(a + b/x^(1/3))^3*a^4/b^6 + 3/2*(a + b/x^(1/3))^2*a^5/b^6

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Fricas [A]  time = 1.46623, size = 45, normalized size = 2.37 \begin{align*} -\frac{7 \, a x + 6 \, b x^{\frac{2}{3}}}{14 \, x^{3}} \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]

-1/14*(7*a*x + 6*b*x^(2/3))/x^3

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Sympy [A]  time = 1.20286, size = 17, normalized size = 0.89 \begin{align*} - \frac{a}{2 x^{2}} - \frac{3 b}{7 x^{\frac{7}{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/(7*x**(7/3))

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Giac [A]  time = 1.13663, size = 20, normalized size = 1.05 \begin{align*} -\frac{7 \, a x^{\frac{1}{3}} + 6 \, b}{14 \, x^{\frac{7}{3}}} \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]

-1/14*(7*a*x^(1/3) + 6*b)/x^(7/3)