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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 19, 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} \[ -\frac {a}{2 x^2}-\frac {3 b}{7 x^{7/3}} \]

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 = 0.01, size = 19, normalized size = 1.00 \[ -\frac {a}{2 x^2}-\frac {3 b}{7 x^{7/3}} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 16, normalized size = 0.84 \[ -\frac {7 \, a x + 6 \, b x^{\frac {2}{3}}}{14 \, x^{3}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 15, normalized size = 0.79 \[ -\frac {7 \, a x^{\frac {1}{3}} + 6 \, b}{14 \, x^{\frac {7}{3}}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ -\frac {a}{2 x^{2}}-\frac {3 b}{7 x^{\frac {7}{3}}} \]

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*a/x^2

________________________________________________________________________________________

maxima [B]  time = 0.52, size = 98, normalized size = 5.16 \[ -\frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7}}{7 \, b^{6}} + \frac {5 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a}{2 \, b^{6}} - \frac {6 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{2}}{b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{3}}{2 \, b^{6}} - \frac {5 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{4}}{b^{6}} + \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{5}}{2 \, b^{6}} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.13, size = 13, normalized size = 0.68 \[ -\frac {a}{2\,x^2}-\frac {3\,b}{7\,x^{7/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^(1/3))/x^3,x)

[Out]

- a/(2*x^2) - (3*b)/(7*x^(7/3))

________________________________________________________________________________________

sympy [A]  time = 1.32, size = 17, normalized size = 0.89 \[ - \frac {a}{2 x^{2}} - \frac {3 b}{7 x^{\frac {7}{3}}} \]

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))

________________________________________________________________________________________