3.24.100 \(\int \frac {a+\frac {b}{\sqrt [3]{x}}}{x^2} \, dx\) [2400]

Optimal. Leaf size=17 \[ -\frac {3 b}{4 x^{4/3}}-\frac {a}{x} \]

[Out]

-3/4*b/x^(4/3)-a/x

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Rubi [A]
time = 0.00, 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} \begin {gather*} -\frac {a}{x}-\frac {3 b}{4 x^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))/x^2,x]

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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^2} \, dx &=\int \left (\frac {b}{x^{7/3}}+\frac {a}{x^2}\right ) \, dx\\ &=-\frac {3 b}{4 x^{4/3}}-\frac {a}{x}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} -\frac {3 b}{4 x^{4/3}}-\frac {a}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))/x^2,x]

[Out]

(-3*b)/(4*x^(4/3)) - a/x

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Maple [A]
time = 0.02, size = 14, normalized size = 0.82

method	result	size
derivativedivides	\(-\frac {3 b}{4 x^{\frac {4}{3}}}-\frac {a}{x}\)	\(14\)
default	\(-\frac {3 b}{4 x^{\frac {4}{3}}}-\frac {a}{x}\)	\(14\)
trager	\(\frac {a \left (x -1\right )}{x}-\frac {3 b}{4 x^{\frac {4}{3}}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))/x^2,x,method=_RETURNVERBOSE)

[Out]

-3/4*b/x^(4/3)-a/x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (13) = 26\).
time = 0.31, size = 47, normalized size = 2.76 \begin {gather*} -\frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4}}{4 \, b^{3}} + \frac {2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a}{b^{3}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{2}}{2 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))/x^2,x, algorithm="maxima")

[Out]

-3/4*(a + b/x^(1/3))^4/b^3 + 2*(a + b/x^(1/3))^3*a/b^3 - 3/2*(a + b/x^(1/3))^2*a^2/b^3

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Fricas [A]
time = 0.39, size = 16, normalized size = 0.94 \begin {gather*} -\frac {4 \, a x + 3 \, b x^{\frac {2}{3}}}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))/x^2,x, algorithm="fricas")

[Out]

-1/4*(4*a*x + 3*b*x^(2/3))/x^2

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Sympy [A]
time = 0.21, size = 14, normalized size = 0.82 \begin {gather*} - \frac {a}{x} - \frac {3 b}{4 x^{\frac {4}{3}}} \end {gather*}

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/(4*x**(4/3))

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Giac [A]
time = 0.50, size = 15, normalized size = 0.88 \begin {gather*} -\frac {4 \, a x^{\frac {1}{3}} + 3 \, b}{4 \, x^{\frac {4}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))/x^2,x, algorithm="giac")

[Out]

-1/4*(4*a*x^(1/3) + 3*b)/x^(4/3)

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Mupad [B]
time = 1.13, size = 13, normalized size = 0.76 \begin {gather*} -\frac {a}{x}-\frac {3\,b}{4\,x^{4/3}} \end {gather*}

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)/(4*x^(4/3))

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