3.617 \(\int \frac{a+b x^4}{x^8} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{7 x^7}-\frac{b}{3 x^3} \]

[Out]

-a/(7*x^7) - b/(3*x^3)

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Rubi [A]  time = 0.0046928, 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} \[ -\frac{a}{7 x^7}-\frac{b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)/x^8,x]

[Out]

-a/(7*x^7) - b/(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+b x^4}{x^8} \, dx &=\int \left (\frac{a}{x^8}+\frac{b}{x^4}\right ) \, dx\\ &=-\frac{a}{7 x^7}-\frac{b}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0019045, size = 17, normalized size = 1. \[ -\frac{a}{7 x^7}-\frac{b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)/x^8,x]

[Out]

-a/(7*x^7) - b/(3*x^3)

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Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{7\,{x}^{7}}}-{\frac{b}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)/x^8,x)

[Out]

-1/7*a/x^7-1/3*b/x^3

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Maxima [A]  time = 0.96919, size = 20, normalized size = 1.18 \begin{align*} -\frac{7 \, b x^{4} + 3 \, a}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^8,x, algorithm="maxima")

[Out]

-1/21*(7*b*x^4 + 3*a)/x^7

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Fricas [A]  time = 1.3969, size = 36, normalized size = 2.12 \begin{align*} -\frac{7 \, b x^{4} + 3 \, a}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^8,x, algorithm="fricas")

[Out]

-1/21*(7*b*x^4 + 3*a)/x^7

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Sympy [A]  time = 0.315085, size = 15, normalized size = 0.88 \begin{align*} - \frac{3 a + 7 b x^{4}}{21 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)/x**8,x)

[Out]

-(3*a + 7*b*x**4)/(21*x**7)

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Giac [A]  time = 1.10169, size = 20, normalized size = 1.18 \begin{align*} -\frac{7 \, b x^{4} + 3 \, a}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^8,x, algorithm="giac")

[Out]

-1/21*(7*b*x^4 + 3*a)/x^7