3.215 \(\int \frac{a+b x^3}{x^6} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{5 x^5}-\frac{b}{2 x^2} \]

[Out]

-a/(5*x^5) - b/(2*x^2)

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Rubi [A]  time = 0.0045445, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ -\frac{a}{5 x^5}-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)/x^6,x]

[Out]

-a/(5*x^5) - b/(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+b x^3}{x^6} \, dx &=\int \left (\frac{a}{x^6}+\frac{b}{x^3}\right ) \, dx\\ &=-\frac{a}{5 x^5}-\frac{b}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0018206, size = 17, normalized size = 1. \[ -\frac{a}{5 x^5}-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)/x^6,x]

[Out]

-a/(5*x^5) - b/(2*x^2)

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Maple [A]  time = 0.005, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{5\,{x}^{5}}}-{\frac{b}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)/x^6,x)

[Out]

-1/5*a/x^5-1/2/x^2*b

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Maxima [A]  time = 0.964486, size = 20, normalized size = 1.18 \begin{align*} -\frac{5 \, b x^{3} + 2 \, a}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)/x^6,x, algorithm="maxima")

[Out]

-1/10*(5*b*x^3 + 2*a)/x^5

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Fricas [A]  time = 1.67182, size = 36, normalized size = 2.12 \begin{align*} -\frac{5 \, b x^{3} + 2 \, a}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)/x^6,x, algorithm="fricas")

[Out]

-1/10*(5*b*x^3 + 2*a)/x^5

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Sympy [A]  time = 0.35123, size = 15, normalized size = 0.88 \begin{align*} - \frac{2 a + 5 b x^{3}}{10 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)/x**6,x)

[Out]

-(2*a + 5*b*x**3)/(10*x**5)

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Giac [A]  time = 1.12995, size = 20, normalized size = 1.18 \begin{align*} -\frac{5 \, b x^{3} + 2 \, a}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)/x^6,x, algorithm="giac")

[Out]

-1/10*(5*b*x^3 + 2*a)/x^5