3.2402 \(\int \frac{a+\frac{b}{\sqrt [3]{x}}}{x^4} \, dx\)

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

[Out]

(-3*b)/(10*x^(10/3)) - a/(3*x^3)

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Rubi [A]  time = 0.0048825, 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} \[ -\frac{a}{3 x^3}-\frac{3 b}{10 x^{10/3}} \]

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*}

Mathematica [A]  time = 0.0079347, size = 19, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{3 b}{10 x^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))/x^4,x]

[Out]

(-3*b)/(10*x^(10/3)) - a/(3*x^3)

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Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \begin{align*} -{\frac{3\,b}{10}{x}^{-{\frac{10}{3}}}}-{\frac{a}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.968687, size = 201, normalized size = 10.58 \begin{align*} -\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{align*}

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 = 1.47682, size = 46, normalized size = 2.42 \begin{align*} -\frac{10 \, a x + 9 \, b x^{\frac{2}{3}}}{30 \, x^{4}} \end{align*}

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 = 2.05283, size = 17, normalized size = 0.89 \begin{align*} - \frac{a}{3 x^{3}} - \frac{3 b}{10 x^{\frac{10}{3}}} \end{align*}

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 = 1.17239, size = 20, normalized size = 1.05 \begin{align*} -\frac{10 \, a x^{\frac{1}{3}} + 9 \, b}{30 \, x^{\frac{10}{3}}} \end{align*}

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)