3.619 \(\int \frac{a+b x^4}{x^{10}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{9 x^9}-\frac{b}{5 x^5} \]

[Out]

-a/(9*x^9) - b/(5*x^5)

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Rubi [A]  time = 0.0047332, 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} \[ -\frac{a}{9 x^9}-\frac{b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)/x^10,x]

[Out]

-a/(9*x^9) - b/(5*x^5)

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^{10}} \, dx &=\int \left (\frac{a}{x^{10}}+\frac{b}{x^6}\right ) \, dx\\ &=-\frac{a}{9 x^9}-\frac{b}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0017928, size = 17, normalized size = 1. \[ -\frac{a}{9 x^9}-\frac{b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)/x^10,x]

[Out]

-a/(9*x^9) - b/(5*x^5)

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Maple [A]  time = 0.004, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{9\,{x}^{9}}}-{\frac{b}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)/x^10,x)

[Out]

-1/9*a/x^9-1/5*b/x^5

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Maxima [A]  time = 0.95868, size = 20, normalized size = 1.18 \begin{align*} -\frac{9 \, b x^{4} + 5 \, a}{45 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^10,x, algorithm="maxima")

[Out]

-1/45*(9*b*x^4 + 5*a)/x^9

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Fricas [A]  time = 1.39998, size = 36, normalized size = 2.12 \begin{align*} -\frac{9 \, b x^{4} + 5 \, a}{45 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^10,x, algorithm="fricas")

[Out]

-1/45*(9*b*x^4 + 5*a)/x^9

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Sympy [A]  time = 0.405909, size = 15, normalized size = 0.88 \begin{align*} - \frac{5 a + 9 b x^{4}}{45 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)/x**10,x)

[Out]

-(5*a + 9*b*x**4)/(45*x**9)

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Giac [A]  time = 1.08955, size = 20, normalized size = 1.18 \begin{align*} -\frac{9 \, b x^{4} + 5 \, a}{45 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^10,x, algorithm="giac")

[Out]

-1/45*(9*b*x^4 + 5*a)/x^9