3.25.2 \(\int \frac {a+\frac {b}{\sqrt [3]{x}}}{x^4} \, dx\) [2402]

Optimal. Leaf size=19 \[ -\frac {3 b}{10 x^{10/3}}-\frac {a}{3 x^3} \]

[Out]

-3/10*b/x^(10/3)-1/3*a/x^3

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Rubi [A]
time = 0.00, 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} \begin {gather*} -\frac {a}{3 x^3}-\frac {3 b}{10 x^{10/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))/x^4,x]

[Out]

(-3*b)/(10*x^(10/3)) - a/(3*x^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+\frac {b}{\sqrt [3]{x}}}{x^4} \, dx &=\int \left (\frac {b}{x^{13/3}}+\frac {a}{x^4}\right ) \, dx\\ &=-\frac {3 b}{10 x^{10/3}}-\frac {a}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.11 \begin {gather*} \frac {-9 b-10 a \sqrt [3]{x}}{30 x^{10/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))/x^4,x]

[Out]

(-9*b - 10*a*x^(1/3))/(30*x^(10/3))

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Maple [A]
time = 0.03, size = 14, normalized size = 0.74

method	result	size
derivativedivides	\(-\frac {3 b}{10 x^{\frac {10}{3}}}-\frac {a}{3 x^{3}}\)	\(14\)
default	\(-\frac {3 b}{10 x^{\frac {10}{3}}}-\frac {a}{3 x^{3}}\)	\(14\)
trager	\(\frac {a \left (x^{2}+x +1\right ) \left (x -1\right )}{3 x^{3}}-\frac {3 b}{10 x^{\frac {10}{3}}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(1/3))/x^4,x,method=_RETURNVERBOSE)

[Out]

-3/10*b/x^(10/3)-1/3*a/x^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (13) = 26\).
time = 0.29, size = 149, normalized size = 7.84 \begin {gather*} -\frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{10}}{10 \, b^{9}} + \frac {8 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{9} a}{3 \, b^{9}} - \frac {21 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{8} a^{2}}{2 \, b^{9}} + \frac {24 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7} a^{3}}{b^{9}} - \frac {35 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a^{4}}{b^{9}} + \frac {168 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{5}}{5 \, b^{9}} - \frac {21 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{6}}{b^{9}} + \frac {8 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{7}}{b^{9}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{8}}{2 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))/x^4,x, algorithm="maxima")

[Out]

-3/10*(a + b/x^(1/3))^10/b^9 + 8/3*(a + b/x^(1/3))^9*a/b^9 - 21/2*(a + b/x^(1/3))^8*a^2/b^9 + 24*(a + b/x^(1/3
))^7*a^3/b^9 - 35*(a + b/x^(1/3))^6*a^4/b^9 + 168/5*(a + b/x^(1/3))^5*a^5/b^9 - 21*(a + b/x^(1/3))^4*a^6/b^9 +
8*(a + b/x^(1/3))^3*a^7/b^9 - 3/2*(a + b/x^(1/3))^2*a^8/b^9

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Fricas [A]
time = 0.38, size = 16, normalized size = 0.84 \begin {gather*} -\frac {10 \, a x + 9 \, b x^{\frac {2}{3}}}{30 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))/x^4,x, algorithm="fricas")

[Out]

-1/30*(10*a*x + 9*b*x^(2/3))/x^4

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Sympy [A]
time = 0.43, size = 17, normalized size = 0.89 \begin {gather*} - \frac {a}{3 x^{3}} - \frac {3 b}{10 x^{\frac {10}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(1/3))/x**4,x)

[Out]

-a/(3*x**3) - 3*b/(10*x**(10/3))

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Giac [A]
time = 0.52, size = 15, normalized size = 0.79 \begin {gather*} -\frac {10 \, a x^{\frac {1}{3}} + 9 \, b}{30 \, x^{\frac {10}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(1/3))/x^4,x, algorithm="giac")

[Out]

-1/30*(10*a*x^(1/3) + 9*b)/x^(10/3)

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Mupad [B]
time = 1.11, size = 13, normalized size = 0.68 \begin {gather*} -\frac {a}{3\,x^3}-\frac {3\,b}{10\,x^{10/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^(1/3))/x^4,x)

[Out]

- a/(3*x^3) - (3*b)/(10*x^(10/3))

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